Calculus - User contributions [en]

MediaWiki:Sitenotice

2024-09-08T03:59:58Z

Vipul:

Want site search autocompletion? See [[Project:Enabling site search autocompletion|here]] 
Encountering 429 Too Many Requests errors when browsing the site? See [[Project:429 Too Many Requests error|here]]

User:Vipul/Sandbox

2024-09-08T03:59:08Z

Vipul:

* <math>9^{\sqrt{7 + 2}} = 729</math>
* <math>(7 + 2)^\sqrt{9} = 729</math>
* <math>6^{2 + 1} = 216</math>
* <math>(3 + 4)^3 = 343</math>

MediaWiki:Sitenotice

2024-09-06T17:28:33Z

Vipul:

'''This site is in the process of being migrated to a new server. Edits made until this notice has been removed may be lost.''' 
Want site search autocompletion? See [[Project:Enabling site search autocompletion|here]] 
Encountering 429 Too Many Requests errors when browsing the site? See [[Project:429 Too Many Requests error|here]]

Calculus:Enabling site search autocompletion

2024-08-07T21:29:38Z

Vipul: /* How to fix it */

Content copied from [[Ref:Ref:Enabling site search autocompletion]]. Images used are specific to this site (Calculus).

Site search autocompletion is currently broken by default on this site. This page includes details on how to get it to work, and what's going on.

==What's wrong with site search autocompletion and how to fix it==

===What's wrong===

When you start typing something in the site search bar, you'll see it stuck at "Loading search suggestions" as shown in the screenshot below:

[[File:Site search autocompletion broken.png]]

Note that the actual search is still working -- you just have to hit Enter after typing the search query and it'll go to the search results page. It's the autocompletion before you hit Enter that is broken.

===How to fix it===

To fix it, you need to follow these steps:

* Write to vipulnaik1@gmail.com asking for a login to the site. Please include the following with your request: preferred username, preferred initial password (you can change it after logging in), real name (if you want it entered), email address to use (if you want an actual email address by which you can be contacted), and whether you want edit access as well. You don't need edit access for enabling site search autocompletion.
* Log in to the site. Then go to [[Special:Preferences]]. Go to the Appearance section and switch the Skin from "Vector (2022)" to "Vector legacy (2010)".
* Make sure to hit "Save" at the bottom.
* Now you can reload the page or load a new page.

Site search autocompletion should now work. Here's an example:

[[File:Site search autocompletion working.png]]

==More background==

We've recently upgraded the MediaWiki version of this wiki from 1.35.13 to 1.41.2 (see [[Special:Version]]). The upgrade allows us to migrate the wiki to a more modern operating system version running PHP 8. With the current setup for MediaWiki 1.41.2, we're in this situation:

* The "Vector legacy (2010)" skin has site search autocompletion working, but it doesn't render well on small screens. Specifically, even on small mobile screens, it still shows the left menu, and doesn't properly use the MobileFrontend extension settings.
* The "Vector (2022)" skin doesn't have site search autocompletion working (see screenshots in preceding section) but it does render fine on mobile devices.

It is possible to set only one default skin (that is applicable to all non-logged-in users and is the default for logged-in users who have not configured a skin for themselves). So, the selection of default skin comes down to whether it's more important for casual users to have the mobile experience working or to have site search autocompletion working. Based on a general understanding of user behavior, we believe that having a usable mobile experience is more important for casual users than having site search autocompletion.

However, for power users who are using the site extensively, site search autocompletion may be important. That's why we've written this page giving guidance on how to set up site search autocompletion.

MediaWiki:Sitenotice

2024-08-07T21:21:48Z

Vipul:

Calculus:429 Too Many Requests error

2024-08-07T21:20:23Z

Vipul: Created page with "This content is copied from Ref:Ref:429 Too Many Requests error. If you get a 429 Too Many Requests error when browsing this site, read on. You're probably seeing this error because a large number of requests have been made from your IP address over a short period of time. That's probably a lot of requests from you or others who share your IP address (such as your home wi-fi network). Waiting a minute and then retrying should generally work. If you are an actual h..."

This content is copied from [[Ref:Ref:429 Too Many Requests error]].

If you get a 429 Too Many Requests error when browsing this site, read on.

You're probably seeing this error because a large number of requests have been made from your IP address over a short period of time. That's probably a lot of requests from you or others who share your IP address (such as your home wi-fi network). Waiting a minute and then retrying should generally work.

If you are an actual human being with a legitimate reason to be browsing the site heavily, first, thank you and sorry about this! We set rate limits to prevent bots, spiders, spammers, and malicious actors from consuming too much of our server's resources so that our server's resources can be devoted to real humans like you. Consider writing to vipulnaik1@gmail.com with your IP address to have the IP address whitelisted. You can get your IP address by [https://www.google.com/search?q=my+ip+address Googling "my IP address"] (scroll down a little bit to where Google includes the IP address in a box). NOTE: If you have both an IPv4 address and an IPv6 address, you should send both; the server supports both IPv4 and IPv6, so either may end up getting used. To check if you have an IPv6 address, try visiting [https://ipv6.google.com/ ipv6.google.com].

If your IP address changes, or you are away from your home network, then you'll get rate-limited again. So if you find yourself getting rate-limited after already having been whitelisted, check if you are on a different IP address than the one for which you requested whitelisting.

File:Site search autocompletion working.png

2024-08-07T21:18:40Z

Vipul:

File:Site search autocompletion broken.png

2024-08-07T21:18:14Z

Vipul:

Calculus:Enabling site search autocompletion

2024-08-07T21:15:50Z

Vipul: Created page with "Content copied from Ref:Ref:Enabling site search autocompletion. Images used are specific to this site (Calculus). Site search autocompletion is currently broken by default on this site. This page includes details on how to get it to work, and what's going on. ==What's wrong with site search autocompletion and how to fix it== ===What's wrong=== When you start typing something in the site search bar, you'll see it stuck at "Loading search suggestions" as shown in..."

Content copied from [[Ref:Ref:Enabling site search autocompletion]]. Images used are specific to this site (Calculus).

Site search autocompletion is currently broken by default on this site. This page includes details on how to get it to work, and what's going on.

==What's wrong with site search autocompletion and how to fix it==

===What's wrong===

When you start typing something in the site search bar, you'll see it stuck at "Loading search suggestions" as shown in the screenshot below:

[[File:Site search autocompletion broken.png]]

Note that the actual search is still working -- you just have to hit Enter after typing the search query and it'll go to the search results page. It's the autocompletion before you hit Enter that is broken.

===How to fix it===

To fix it, you need to follow these steps:

* Write to vipulnaik1@gmail.com asking for a login to the site. Please include the following with your request: preferred username, preferred initial password (you can change it after logging in), real name (if you want it entered), email address to use (if you want an actual email address by which you can be contacted), and whether you want edit access as well. You don't need edit access for enabling site search autocompletion.
* Log in to the site, and go to [[Special:Preferences]]. Go to the Appearance section and switch the Skin from "Vector (2022)" to "Vector legacy (2010)".
* Make sure to hit "Save" at the bottom.
* Now you can reload the page or load a new page.

Site search autocompletion should now work. Here's an example:

[[File:Site search autocompletion working.png]]

==More background==

We've recently upgraded the MediaWiki version of this wiki from 1.35.13 to 1.41.2 (see [[Special:Version]]). The upgrade allows us to migrate the wiki to a more modern operating system version running PHP 8. With the current setup for MediaWiki 1.41.2, we're in this situation:

* The "Vector legacy (2010)" skin has site search autocompletion working, but it doesn't render well on small screens. Specifically, even on small mobile screens, it still shows the left menu, and doesn't properly use the MobileFrontend extension settings.
* The "Vector (2022)" skin doesn't have site search autocompletion working (see screenshots in preceding section) but it does render fine on mobile devices.

It is possible to set only one default skin (that is applicable to all non-logged-in users and is the default for logged-in users who have not configured a skin for themselves). So, the selection of default skin comes down to whether it's more important for casual users to have the mobile experience working or to have site search autocompletion working. Based on a general understanding of user behavior, we believe that having a usable mobile experience is more important for casual users than having site search autocompletion.

However, for power users who are using the site extensively, site search autocompletion may be important. That's why we've written this page giving guidance on how to set up site search autocompletion.

MediaWiki:Sitenotice

2024-08-07T21:14:36Z

Vipul: Blanked the page

User:Vipul/Sandbox

2024-08-07T21:10:36Z

Vipul:

* <math>9^{\sqrt{7 + 2}} = 729</math>
* <math>(7 + 2)^\sqrt{9} = 729</math>
* <math>6^{2 + 1} = 216</math>

User:Vipul/Sandbox

2024-08-07T21:03:45Z

Vipul:

* <math>9^{\sqrt{7 + 2}} = 729</math>
* <math>(7 + 2)^\sqrt{9} = 729</math>

User:Vipul/Sandbox

2024-08-07T20:58:59Z

Vipul: Created page with "* <math>9^{\sqrt{7 + 2}} = 729</math>"

* <math>9^{\sqrt{7 + 2}} = 729</math>

MediaWiki:Sitenotice

2024-08-07T20:42:31Z

Vipul: Created page with "'''This wiki is in the process of being upgraded. The site may go down intermittently. Please try to avoid editing until this notice has been removed.'''"

'''This wiki is in the process of being upgraded. The site may go down intermittently. Please try to avoid editing until this notice has been removed.'''

Point of local extremum implies critical point

2024-05-13T06:12:28Z

Vipul: /* General comments */

{{maxmin test}}
==Statement==

Suppose <math>f</math> is a [[function]] of one variable and <math>c</math> is a point in the [[interior]] of the [[domain]] of <math>f</math> (i.e., <math>f</math> is defined on an open interval containing <math>c</math>).

Suppose further that <math>c</math> is a [[point of local extremum]] for <math>f</math>, i.e., <math>f</math> attains a local extreme value (either a local maximum or a local minimum) at <math>c</math>.

Then, <math>c</math> is a [[critical point]] for <math>f</math>, i.e., either the [[derivative]] <math>f'(c)</math> equals zero or the derivative <math>f'(c)</math> does not exist.

Note that the <math>f'(c)</math> not existing case could occur in either of these ways: one or both the one-sided derivatives of <math>f</math> at <math>c</math> not existing, ''or'' both one-sided derivatives of <math>f</math> at <math>c</math> existing but being unequal.

==Related facts==

* [[Point of local extremum implies critical point for a function of multiple variables]]

==Significance==

===General comments===

The main significance of this result is that it mostly reduces an ''optimization'' problem (finding a maximum or minimum) to an ''equation-solving'' problem (finding when an expression is zero).

The reduction isn't perfect, in that it has the following caveats:

* The result is only a one-sided implication. Any point of local extremum is a critical point, but it's possible for a critical point to not be a point of local extremum. Even so, in many cases, the set of critical points is already small enough and manageable enough.
* This result is about local extrema, not absolute / global extrema. There may be many points of local extremum that aren't points of absolute extremum. Moreover, there could be many cases where an absolute extremum doesn't exist; for instance, for a function that is going to infinity as the input goes to infinity or to a point of discontinuity (such as <math>x \mapsto 1/x</math> as <math>x \to 0^+</math> and <math>x \to 0^-</math>. With that being said, in many cases, we can rule out limiting behaviors of that sort, and the set of points of local extremum is small enough that it's feasible to find the absolute extremum by looking among them.

===Qualitative and existential significance===

{| class="sortable" border="1"
! Type of function for which we are interested in understanding local extrema !! Maximum number of points of local extremum !! Inference !! Example
|-
| polynomial function of degree <math>d</math> || <math>d - 1</math> || The derivative is a polynomial of degree <math>d - 1</math>, so it is defined everywhere and has at most <math>d - 1</math> real roots. Therefore, there are at most <math>d - 1</math> critical points (all of the "derivative equals zero" type, none of the "derivative is undefined" type) and therefore at most <math>d - 1</math> points of local extremum for the function. || <math>x^3 - 3x</math> of degree <math>d = 3</math> has <math>3 - 1 = 2</math> critical points: the two solutions to <math>3x^2 - 3 = 0</math>, namely <math>x = -1</math> and <math>x = 1</math>. Both of these turn out to be points of local extremum.
|-
| rational function in simplified form with numerator degree <math>d_1</math> and denominator degree <math>d_2</math> || <math>d_1 + d_2 - 1</math> || The derivative is a rational function with numerator degree at most <math>d_1 + d_2 - 1</math> and denominator a square of the original denominator. The only way for the derivative to be undefined is for the denominator to be zero, in which case the denominator of the original rational function would also be zero, so that the original rational function would also be undefined. Therefore, the derivative exists everywhere that the function is defined, and the only critical points are cases where the derivative is zero. Since the numerator is a polynomial of degree at most <math>d_1 + d_2 - 1</math>, there are therefore at most <math>d_1 + d_2 - 1</math> critical points, and therefore at most <math>d_1 + d_2 - 1</math> points of local extremum. || <math>x/(x^2 + 1)</math> with <math>d_1 = 1, d_2 = 2</math> has derivative <math>(1 - x^2)/(x^2 + 1)^2</math> and has <math>1 + 2 - 1 = 2</math> critical points, namely the solutions to <math>1 - x^2 = 0</math>, so <math>x = -1</math> and <math>x = 1</math>. Both of these turn out to be points of local extremum.
|}

===Computational feasibility significance===

As a general rule, for any family of functions <math>\mathcal{F}</math> for which we know how to find the roots of any function in the family (i.e., the points where the function is zero), we have a strategy to find critical points (and therefore points of local extremum) for any function whose ''derivative'' is in <math>\mathcal{F}</math>.

In particular, since we know how to find the roots of polynomials of degree 1 (linear polynomial) or 2 (quadratic polynomial), we can find the critical points of any degree 2 (quadratic) or degree 3 (cubic) function, as well as of any rational function with a linear numerator and quadratic denominator.

From the perspective of numerical approximation, any method for numerically approximating the ''roots'' of a function in a family <math>\mathcal{F}</math> can become a method for finding the critical points (and hence, the points of local extremum) of a function whose derivative is in <math>\mathcal{F}</math>. Some of these translations are below:

{| class="sortable" border="1"
! Root-finding method !! Corresponding optimization method
|-
| [[Newton's method for root-finding for a function of one variable]] || [[Newton's method for optimization of a function of one variable]]
|}

==Facts used==

===Statement of facts===

{| class="sortable" border="1"
! Fact no. (for reference in proof) !! Statement !! Assumption about one-sided local extremum !! Conclusion about sign of one-sided derivative !! Quick explanation in terms of the [[difference quotient]] <math>\frac{f(x) - f(c)}{x - c}</math> !! Prototypical pictures
|-
| 1 || [[uses::Local maximum from the left implies left-hand derivative is nonnegative if it exists]] (has full proof + video) || <math>f</math> has a (possibly, but not necessarily, strict) local maximum from the left at <math>c</math>. || The left-hand derivative <math>f'_-(c)</math>, if it exists, is positive or zero || In the difference quotient, the numerator is negative or zero and the denominator is negative. The quotient is thus positive or zero. Hence, so is the limit, if it exists. || [[File:Leftincreasingconcaveup.png|100px]][[File:Leftincreasingconcavedownflat.png|100px]][[File:Leftincreasingoscillatoryflat.png|100px]]
|-
| 2 || [[uses::Local minimum from the left implies left-hand derivative is nonpositive if it exists]] || <math>f</math> has a (possibly, but not necessarily, strict) local minimum from the left at <math>c</math>. || The left-hand derivative <math>f'_-(c)</math>, if it exists, is negative or zero || In the difference quotient, the numerator is positive or zero and the denominator is negative. The quotient is thus negative or zero. Hence, so is the limit, if it exists. || [[File:Leftdecreasingconcavedown.png|100px]][[File:Leftdecreasingconcaveupflat.png|100px]]
|-
| 3 || [[uses::Local maximum from the right implies right-hand derivative is nonpositive if it exists]] || <math>f</math> has a (possibly, but not necessarily, strict) local maximum from the right at <math>c</math>. || The right-hand derivative <math>f'_+(c)</math>, if it exists, is negative or zero || In the difference quotient, the numerator is negative or zero and the denominator is positive. The quotient is thus negative or zero. Hence, so is the limit, if it exists. || [[File:Rightdecreasingconcavedownnotflat.png|100px]][[File:Rightdecreasingconcaveup.png|100px]]
|-
| 4 ||[[uses::Local minimum from the right implies right-hand derivative is nonnegative if it exists]] || <math>f</math> has a (possibly, but not necessarily, strict) local minimum from the right at <math>c</math>. || The right-hand derivative <math>f'_+(c)</math>, if it exists, is positive or zero || In the difference quotient, the numerator is positive or zero and the denominator is positive. The quotient is thus positive or zero. Hence, so is the limit, if it exists. || [[File:Rightincreasingconcavedown.png|100px]][[File:Rightincreasingconcaveupnotflat.png|100px]]
|-
|}

The video below provides an intuitive explanation of the above facts. For a full proof, see the page on Fact (1).

<center>{{#widget:YouTube|id=sko6usPekeQ}}</center>

===Note on strictness===

All the facts as stated above are for not necessarily strict one-sided local extrema, i.e., we allow <math>f(x) = f(c)</math> for <math>x</math> on the immediate left or right of <math>c</math>. However, even if we impose the additional condition of strictness, we still cannot eliminate the possibility of the one-sided derivative being zero. The reason is that even though the difference quotient must now be strictly positive or strictly negative in the various cases, the one-sided derivative, which is defined as a one-sided limit of the difference quotient, may still be equal to zero. All the prototypical pictures in the previous section are pictures of strict local extrema, and some of them show a one-sided derivative of zero.

For more, see the note on strictness in the proof for Fact (1).

This observation will be crucial when we piece together the two-sided information.

===Note on sign sensitivity===

In the facts used above, we see that, when ascertaining the sign of the [[difference quotient]] <math>\frac{f(x) - f(c)}{x - c}</math>, the following are true:

* The sign of the numerator is governed by whether the point is a point of local maximum or minimum. For a point of local maximum, the numerator is negative or zero, and for a point of local minimum, the numerator is positive or zero.
* The sign of the denominator is governed by whether we are approaching from the left or the right. For a left-sided approach, the denominator is negative and for a right-sided approach, the denominator is positive.

The upshot of this is that if we change the direction of approach while preserving the nature of the local extreme value, the sign of the one-sided derivative flips. This is crucial to the proof.

==Proof==

===Proof idea===

The idea is to convert ''two-sided'' local extremum information into both its one-sided pieces, then determine the signs of the one-sided derivatives. As noted in [[#Note on sign sensitivity]], the sign conclusions for the two one-sided derivatives are opposite. We then pit these two pieces of information against each other to force the two-sided derivative, if it exists, to equal zero.

===Local maximum case===

'''Given''': A function <math>f</math>, a point <math>c</math> in the interior of the domain of <math>f</math> such that <math>f</math> attains a local maximum at <math>c</math>, i.e., <math>f(x) \le f(c)</math> for all <math>x \in (c - \delta, c + \delta)</math> for some choice of <math>\delta > 0</math>.

'''To prove''': If <math>\! f'(c)</math> exists, then <math>\!f'(c) = 0</math>

'''Proof'''

{| class="sortable" border="1"
! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation
|-
| 1 || If the left-hand derivative of <math>f</math> at <math>c</math> exists, then it is nonnegative, i.e., it is positive or zero. || Fact (1) || <math>f</math> attains a (two-sided) local maximum at <math>c</math> || || Since <math>f</math> attains a ''two-sided'' local maximum at <math>c</math>, it in particular attains a local maximum from the left. Thus, Fact (1) applies.
|-
| 2 || If the right-hand derivative of <math>f</math> at <math>c</math> exists, then it is nonpositive, i.e., it is negative or zero. || Fact (3) || <math>f</math> attains a (two-sided) local maximum at <math>c</math> || || Since <math>f</math> attains a ''two-sided'' local maximum at <math>c</math>, it in particular attains a local maximum from the right. Thus, Fact (3) applies.
|-
| 3 || If the (two-sided) derivative of <math>f</math> exists at <math>c</math>, it must be zero. || || || Steps (1), (2) || For the (two-sided) derivative to exist, ''both'' the left-hand derivative and the right-hand derivative must exist ''and'' they must be equal to each other and to the derivative. By Step (1), the derivative must be positive or zero. By Step (2), the derivative must be negative or zero. The only way both of these can be simultaneously true is if the derivative equals zero.
|}

<center>{{#widget:YouTube|id=gExgdK8xKUs}}</center>

===Local minimum case===

'''Given''': A function <math>f</math>, a point <math>c</math> in the interior of the domain of <math>f</math> such that <math>f</math> attains a local minimum at <math>c</math>, i.e., <math>f(x) \ge f(c)</math> for all <math>x \in (c - \delta, c + \delta)</math> for some choice of <math>\delta > 0</math>.

'''To prove''': If <math>f'(c)</math> exists, then <math>f'(c) = 0</math>

'''Proof'''

{| class="sortable" border="1"
! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation
|-
| 1 || If the left-hand derivative of <math>f</math> at <math>c</math> exists, then it is nonpositive, i.e., it is negative or zero. || Fact (2) || <math>f</math> attains a (two-sided) local minimum at <math>c</math> || || Since <math>f</math> attains a ''two-sided'' local minimum at <math>c</math>, it in particular attains a local minimum from the left. Thus, Fact (2) applies.
|-
| 2 || If the right-hand derivative of <math>f</math> at <math>c</math> exists, then it is nonnegative, i.e., it is positive or zero. || Fact (4) || <math>f</math> attains a (two-sided) local minimum at <math>c</math> || || Since <math>f</math> attains a ''two-sided'' local minimum at <math>c</math>, it in particular attains a local minimum from the right. Thus, Fact (4) applies.
|-
| 3 || If the (two-sided) derivative of <math>f</math> exists at <math>c</math>, it must be zero. || || || Steps (1), (2) || For the (two-sided) derivative to exist, ''both'' the left-hand derivative and the right-hand derivative must exist ''and'' they must be equal to each other and to the derivative. By Step (1), the derivative must be negative or zero. By Step (2), the derivative must be positive or zero. The only way both of these can be simultaneously true is if the derivative equals zero.
|}

===Note on strictness===

Even if we consider the case of ''strict'' two-sided local maximum or strict two-sided local minimum, we can still have either of the two types of critical point: <math>f'(c) = 0</math> or <math>f'(c)</math> not existing. The <math>f'(c) = 0</math> case continues to be possible because, for the one-sided versions, we can have the one-sided derivative equal to zero even assuming ''strict'' one-sided local extremum.

Conversion of a differential equation to a system of first-order differential equations

2024-04-18T07:06:53Z

Vipul:

==Statement==

Consider a [[differential equation]] with independent variable <math>x</math> and dependent variable <math>y</math> given as follows:

<math>F(x,y,y',\dots,y^{(k)}) = 0</math>

We can convert this to a [[system of first-order differential equations]] with <math>k</math> variables and <math>k</math> equations as follows.

Take the variables to be <math>y,y_1,y_2,\dots,y_{k-1}</math> where <math>y_i</math> is the variable that we will eventually set as <math>y^{(i)}</math>. The system has <math>k - 1</math> first-order differential equations of the form:

<math>y' = y_1, y_1' = y_2, \dots, y_{k-2}' = y_{k-1}</math>

and one more first-order differential equation:

<math>F(x,y,y_1,\dots,y_{k-1},y_{k-1}') = 0</math>

Note that, with the exception of the last differential equation, all the other equations are autonomous (no reference to the dependent variable), first-order (the highest derivative order is one) and linear.

Differential equation

2024-04-18T06:56:43Z

Vipul: /* Terminology */

==Definition==

===Formal description===

The term '''differential equation''', sometimes called '''ordinary differential equation''' to distinguish it from [[partial differential equation]]s and other variants, is an equation involving two variables, an ''independent variable'' <math>x</math> and a ''dependent'' variable <math>y</math>, as well as the [[derivative]]s (first and possibly higher) of <math>y</math> with respect to <math>x</math>. Formally, it is an equation of the form:

<math>F(x,y,y',y'',\dots,y^{(k)}) = 0</math>

where <math>F</math> is a function of <math>k + 2</math> variables. Here <math>k \ge 1</math>. Note that <math>F</math> may choose not to use some of the derivatives.

In functional notation, the same differential equation may be written as:

<math>F(x,f(x),f'(x),f''(x),\dots,f^{(k)}(x)) = 0</math>

where <math>f</math> is the function such that <math>y = f(x)</math>.

===Solution concept===

* '''Functional solution''': A function <math>f</math> on the domain of interest is said to be a ''solution'' (or ''functional solution'') to the equation if, when we plug in <math>y = f(x)</math>, the equation holds true for ''all'' <math>x</math> in the domain, i.e.:

<math>F(x,f(x),f'(x),f''(x),\dots,f^{(k)}(x)) = 0 \ \forall \ x \in \operatorname{dom}(f)</math>

Note that in cases of functions defined on closed intervals, we exclude checking the conditions on the boundary of the domain because two-sided derivatives don't make sense at the boundary.

* '''Relational solution''': A relation <math>R(x,y) = 0</math> is termed a ''relational solution'' to the equation if <math>F(x,y,y',y'',\dots,y^{(k)}) = 0</math> holds true for all <math>x,y</math> if we calculate the derivatives of <math>y</math> with respect to <math>x</math> using [[implicit differentiation]].

===Initial value problem===

An ''initial value problem'' is a differential equation:

<math>F(x,y,y',y'',\dots,y^{(k)}) = 0</math>

accompanied with a tuple <math>(x_0,y_0,y_1,\dots,y_{k-1})</math>.

A '''functional solution''' to the initial value problem is a functional solution <math>y = f(x)</math> for the differential equation such that <math>f(x_0) = y_0</math> and <math>f^{(i)}(x_0) = y_i</math> for <math>i \in \{ 1,2,\dots,k-1\}</math>.

Analogously, we can define a relational solution to the initial value problem.

==Key observations==

===Differential equations are functional equations===

Differential equations are examples of [[functional equation]]s. A functional equation is an equation where the ''variable'' that we are trying to solve for is a function, and the equation holds true for all values of the input to the function. For instance, here is an example of a functional equation (that's not a differential equation):

<math>f(x + y) = f(x) + f(y) \ \forall \ x,y \in \R</math>

A ''solution'' to a functional equation is a function that satisfies the equation for all choices of inputs. For instance, any function of the form <math>f(x) := ax</math> for fixed <math>a \in \R</math> is a solution to the above functional equation.

Differential equations are functional equations -- we are trying to solve a differential equation, not for the variables, but for the ''functional'' relationship between them.

===Differential equations capture behavior at a single point===

Not every functional equation involving derivatives is a differential equation. Differential equations are characterized by the evaluation of the function and its derivatives all happening at a single point. For instance:

* <math>x^2 = f(x) + (f'(x))^3</math> is a differential equation because all the function and derivative evaluations happen at <math>x</math>, but
* <math>x^2 = f(x) + f'(1 - x)</math> is ''not'' a differential equation in our sense of the word because the derivative evaluation happens at <math>1 - x</math> rather than <math>x</math>.

Another way of putting this is that differential equations are inherently local and cannot relate the behavior of the function at far-away points.

Functional equations involving derivatives that do ''not'' fit this definition of differential equation are also studied, but the study of these is more complicated and requires new techniques. [[Delay differential equation]]s is one such class of functional equations.

===Functional equations involving iterated derivatives at a single point can be simplied to differential equations===

Consider an equation like this:

<math>(yy''\sin(ye^x + xy'))' = yx^3 + x\cos y</math>

Here, all derivatives are with respect to <math>x</math>.

This does not meet the standard format of a differential equation, because it involves taking a derivative of an intermediate expression <math>yy''\sin(ye^x + xy')</math>, whereas a standard format differential equation only allows for derivatives (including repeated derivatives) of the dependent variable <math>y</math>.

However, this can be ''simplified'' to a differential equation in the standard format, specifically by using various differentiation rules such as the product rule, chain rule, and the linearity of differentiation, plus the formulas for differentiating the sine and exponential functions. Therefore, we generally consider this sort of equation to be an ordinary differential equation, albeit in an unsimplified form.

More abstractly, if <math>G</math> is a function of <math>x, y, y', \dots, y^{(k)}</math>, then <math>G'</math> can be written as follows using [[partial derivative]]s by using the [[chain rule for partial differentiation]]:

<math>G' = G_x + G_y y' + G_{y'} y'' + \dots G_{y^{(k)}} y^{(k+1)}</math>

In particular, this allows us to simplify the intermediate derivative expressions into expressions in <math>x, y, y', \dots, y^{(k + 1)}</math> and thereby allows us to convert the differential equation to a standard format.

(Note that in any particular case such as the example above, we don't need to think in terms of partial derivatives, but the abstract formulation does require the use of partial derivatives).

===It does not make sense to ask whether a point satisfies a differential equation===

Consider a differential equation:

<math>x^2 = y + y'</math>

If I ask the question: ''does the point <math>x = 2,y = 3</math> satisfy the differential equation?'', the answer is that the question doesn't make any sense. This is because verifying a differential equation requires knowing the ''functional relationship'' between <math>x</math> and <math>y</matH>, which in turn allows us to compute the numerical value of <math>y'</math> and check whether the equation is satisfied.

==Terminology==

===Equation terminology===

{| class="sortable" border="1"
! Term !! Meaning !! Example (don't try to solve these differential equations!)
|-
| [[order of a differential equation]] || it is the largest <math>k</math> for which the <math>k^{th}</math> derivative of the dependent variable appears in the differential equation. || The equation <math>y + xy''' + (y'')^2 = \sin(y')</math> has order three because <math>y'''</math> is the largest derivative appearing.
|-
| [[first-order differential equation]] || differential equation of order one, i.e., it involves only <math>x,y,y'</math>. || <math>y' = \sin(x + yy')</math> is a first-order differential equation.
|-
| [[second-order differential equation]] || differential equation of order exactly two, i.e., it involves only <math>x,y,y',y''</math> and has at least one appearance of <math>y''</math>. || <math>xy'' + \cos(y^2y') = x^2e^{yy'}</math> is a second-order differential equation.
|-
| [[degree of a differential equation]] || if the differential equation is polynomial in terms of its highest order derivative, then the degree of that polynomial. || <math>(y''')^2y' + (y')^5 = 3xy</math> has degree two.
|-
| [[explicit differential equation]] || This means that the highest order derivative is written explicitly in terms of the dependent variable, independent variable, and the lower order derivatives. Any explicit differential equation is a first-degree differential equation. Conversely, any first-degree differential equation can be converted to an explicit differential equation by dividing out by the coefficient of the highest order derivative -- if this coefficient is not invertible, we may separately need to consider the case where that coefficient becomes zero, and that would be a ''lower'' order differential equation. || <math>y''' = xy'' - x^2\sin(yy') + y^3(y'')^5</math> is explicit: the third derivative is written in terms of the lower order derivatives.
|-
| [[autonomous differential equation]] || differential equation where the independent variable does ''not'' appear explicitly anywhere in the equation. || <math>y + y'' = \cos(yy'y''')</math> is autonomous. On the other hand, <math>y + xy' = y''</math> is ''not'' autonomous
|-
| [[linear differential equation]] || A differential equation of the form <math>p_k(x)y^{(k)} + p_{k-1}(x)y^{(k-1)} + \dots + p_0(x)y = q(x)</math> where the <math>p_i</math>s and <math>q</math> are all functions. In other words, the expression is linear in <math>y</math> and its derivatives with coefficients in terms of <math>x</math>. Linear differential equations are usually written with the coefficient of <math>y^{(k)}</math> cleared to 1, by dividing throughout by the coefficient of <math>y^{(k)}</math>. || <math>e^xy''' + x^2y'' + \sin(x)y' + 3y = x^2 - 2x + 5</math> is linear.
|-
| [[homogeneous linear differential equation]] || A linear differential equation of the form <math>p_k(x)y^{(k)} + p_{k-1}(x)y^{(k-1)} + \dots + p_0(x)y = 0</math>. In other words, the ''constant term'' function is zero. || <math>e^xy^{(4)} - 3x^3y''' + \sin(\sin x)y = 0</math> is homogeneous linear.
|-
| [[linear differential equation with constant coefficients]] || A linear differential equation of the form <math>a_ky^{(k)} + a_{k-1}y^{(k-1)} + \dots + a_1y' + a_0y = b</math> where all the <math>a_i</math>s and <math>b</math> are zero. || <math>2y^{(5)} - y^{(3)} + y = 13</math> is linear with constant coefficients.
|}

===Solution terminology===

{| class="sortable" border="1"
! Term !! Meaning !! Example
|-
| particular solution || a function or relation that is a solution for the equation (see [[#Solution concept]]). A solution in the form of a function <math>y = f(x)</math> is termed a ''functional solution'' and a solution in the form of a relation <math>R(x,y) = 0</math> is termed a ''relational solution''. || <math>y = \sin x</math> is a functional solution to <math>y^2 + y'^2 = 1</math>.
|-
| solution family || a family of functions or relations, with one or more parameters possibly subject to some constraints, such that for every choice of parameter values subject to those constraints, we get a particular solution. || <math>y = \sin(x + C)</math> with parameter <math>C \in \R</math>, is a solution family for <math>y^2 + y'^2 = 1</math>.
|-
| general solution || a solution family that covers ''all'' solutions (or almost all solutions, possibly excluding some exceptions) || The general solution to <math>y' = 0</math> is <math>y = C, C \in \R</math>.
|-
| solution to initial value problem || a particular solution that satisfies the initial value condition. || A particular solution to <math>y + y' + y'' = (x + 1)^2</math> satisfying <math>y(0) = -1, y'(0) = 0</math> (i.e., <math>y = -1</math> and <math>y' = 0</math> when <math>x = 0</math>) is <math>y = x^2 - 1</math>.
|}

===Variations===

{| class="sortable" border="1"
! Variation name !! How it differs !! Modified name for usual notion to emphasize it's the original notion and not the variation
|-
| [[partial differential equation]] || multiple independent variables, and the use of [[partial derivative]]s instead of ordinary derivatives || ordinary differential equation
|-
| [[system of differential equations]] || multiple dependent variables (but still just one independent variable) and multiple equations relating the dependent variables and their derivatives with each other and the independent variable ||
|}

==Facts==

* As a general principle, the way to solve a differential equation of order <math>k</math> is to reduce it to a sequence of <math>k</math> integration problems. Each integration problem introduces a new freely varying parameter.
* As a general principle, the number of degrees of freedom (i.e., the number of independent freely varying parameters) in the general solution to a differential equation of order <math>k</math> must equal <math>k</math>. There are various exceptions and irregularities, but this is what we should generally expect. Another way of putting this is that the solution space to a differential equation of order <math>k</math> is expected to be <math>k</math>-dimensional.
* As a general principle, the number of solutions to an initial value problem should be finite. If the differential equation is nice enough, then there should be a ''unique'' solution to any initial value problem.

==Relation with system of first-order differential equations==

Any differential equation of order <math>k</math> can be converted to a [[system of first-order differential equations]] with <math>k</math> equations and <math>k</math> variables (i.e., <math>k</math> unknown functions that we are trying to solve for). For the conversion procedure, see [[conversion of a differential equation to a system of first-order differential equations]]. However, the converse is not true, i.e., it is not always possible to convert a system of first-order differential equations with multiple dependent variables into a single differential equation of higher order with one dependent variable.

The same idea can be used to perform the [[conversion of a system of differential equations to a system of first-order differential equations]].

==Solution strategies==

===General idea for strategy toward a general solution===

The rough idea is to convert the differential equation to a sequence of integration problems. If the differential equation has order <math>k</math>, it should reduce to performing <math>k</math> integrations. Each integration introduces a new freely varying parameter.

{{quotation|'''CAUTION''': There may be other auxiliary integrations that need to be done, e.g., for computation of integrating factors or in order to solve an [[integration by parts]] problem. These auxiliary antiderivative computations do not, however, introduce new freely varying parameters.}}

We see that this general strategy may run into trouble at many levels:

{| class="sortable" border="1"
! Stage !! Type of difficulty
|-
| Differential equation to integration || There is no general-purpose algorithm for converting an arbitrary differential equation to an integration problem or sequence of integration problems. Thus, we may not even be able to get started on the process. For some types of structures of differential equations, strategies are known for converting them to integration problems. For others, we have to use ''ad hoc'' techniques. There exist differential equations for which there is no way of converting them to integration problems. Note that for differential equations of order two or higher, this problem may occur at ''any'' of the stages, i.e., we may be able to do one level of integration but not the next.
|-
| Solving the integration problem to get rid of the integral sign || Even after we've converted the differential equation to an integration problem, there may not be any analytic methods for computing the antiderivative. Note that this is not such a big issue because there are known techniques for calculating approximate solutions to integration problems even if analytical methods are not available for getting a precise solution.
|-
| Making sense of relational solutions, converting to functional solutions or at least trying to understand what they mean || Even after we have done the desired antidifferentiations, the solutions may be in relational form rather than functional form. This means that we may still not have ''explicit'' functional descriptions of <math>y</math> in terms of <math>x</math>. We may not even know whether <math>y</math> is expressible as a function of <math>x</math>. We may not even know if the relational solution has any points in it at all!
|}

Historically, this type of method of solution of a differential equation is called a ''solution by quadratures''. In the early days of differential equations, it was hoped that generic differential equations could be solved by quadratures, but this hope was dashed fairly quickly.

===Going from a particular solution to a general solution===

There are some special types of differential equations where, once we find a particular solution, we can find other solutions, perhaps even the general solution. Some notable cases are considered here:

* For a [[linear differential equation]], finding a general solution is equivalent to finding a particular solution + solving the corresponding [[homogeneous linear differential equation]]. In particular, for a [[linear differential equation with constant coefficients]], there is a closed form expression for the solution of the corresponding homogeneous linear differential equation with constant coefficients, so finding a particular solution is equivalent to finding the general solution.
* For an [[autonomous differential equation]], if <math>y = f(x)</math> is a solution, so is <math>y = f(x + C)</math> for any constant <math>C</math>. Note that this does not give the ''general solution'' if the order is more than one, but it does help move from a particular solution to a solution family with one parameter.

===Solution strategies in particular cases===

Below are some formats of equations for which general strategies are known. Note that the letter <math>f</math> is no longer used for the solution function but may be used for other functions.:

{| class="sortable" border="1"
! Equation type !! Order !! Degree (if polynomial in highest order derivative) !! Quick summary of solution strategy
|-
| [[first-order linear differential equation]] which in simplified form looks like <math>y' + p(x)y = q(x)</math> || 1 || 1 || Use the [[integrating factor]] <math>e^{H(x)}</math> where <math>H'=p</math>. The general solution is <math>y = Ce^{-H(x)} + e^{-H(x)}\int p(x)e^{H(x)} \, dx</math>
|-
| [[separable differential equation]] which is of the form <math>y' = f(x)g(y)</math> (any [[first-order first-degree autonomous differential equation]] is separable, though there are separable differential equations that aren't autonomous) || 1 || 1 || Separate and solve as <math>\int \frac{dy}{g(y)} = \int f(x) \, dx</math>. Also find solutions corresponding to <math>y = k</math> where <math>g(k) = 0</math>.
|-
| [[first-order exact differential equation]] <math>F(x,y,y') = 0</math> || 1 || 1 || Try to find a relation <math>R(x,y)</math> such that <math>F(x,y,y') = \frac{d}{dx}[R(x,y)]</math> using [[implicit differentiation]]. Finding the <math>R</math>, even if it does exist, can be tricky.
|-
| [[Bernoulli differential equation]] <math>y' + p(x)y = q(x)y^n</math> (<math>n \ne 0,1</math>) || 1 || 1 || Divide both sides by <math>y^n</math> (set aside possible stationary solution <math>y = 0</math>), then substitute <math>w = 1/y^{n-1}</math> to get a [[first-order linear differential equation]] with dependent variable <math>w</math> and independent variable <math>x</math>.
|-
| [[Clairaut's equation]] which is of the form <math>y = xy' + f(y')</math> || 1 || need not be polynomial; if polynomial, may have any degree || <math>y = Cx + f(C)</math> with <math>C \in \R</math> (all straight lines) and a single other solution explicitly described as the solution to <math>x + f(dy/dx) = 0</math>, given by <math>x = -f'(p), y = f(p) - pf'(p)</math> as a parametric curve in terms of <math>p</math>.
|-
| [[Lagrange equation]] <math>y = f(y')x + g(y')</math> which is linear in <math>x</math> and <math>y</math> but not necessarily in <math>y'</math> || 1 || need not be polynomial; if polynomial, may have any degree || General solution is a family of curves, each described as a parametric curve with parameter the derivative <math>y'</math> (which we denote by <math>p</math>). There may be some special straight line solutions of the form <math>y = px + g(p)</math> for values <math>p</math> with <math>p = f(p)</math>.
|-
| [[second-order autonomous differential equation of degree one]], which is of the form <math>y'' = F(y,y')</math> || 2 || 1 ||
|-
| [[homogeneous linear differential equation with constant coefficients]] || any || 1 || We construct the characteristic polynomial of the differential equation, find its real and complex roots, and the space of solution functions is a vector space with basis functions described using these roots.
|}

===Qualitative methods===

For most differential equations, it is very hard to convert the differential equation to a series of integration problems and to find explicit expressions for the solution. Instead, in many cases, we try to determine the qualitative properties of solution functions, including existence, uniqueness, extent of differentiability, nature of roots and critical points, etc.

Point of local extremum implies critical point

2024-04-18T06:49:36Z

Vipul: /* General comments */

{{maxmin test}}
==Statement==

Suppose <math>f</math> is a [[function]] of one variable and <math>c</math> is a point in the [[interior]] of the [[domain]] of <math>f</math> (i.e., <math>f</math> is defined on an open interval containing <math>c</math>).

Suppose further that <math>c</math> is a [[point of local extremum]] for <math>f</math>, i.e., <math>f</math> attains a local extreme value (either a local maximum or a local minimum) at <math>c</math>.

Then, <math>c</math> is a [[critical point]] for <math>f</math>, i.e., either the [[derivative]] <math>f'(c)</math> equals zero or the derivative <math>f'(c)</math> does not exist.

Note that the <math>f'(c)</math> not existing case could occur in either of these ways: one or both the one-sided derivatives of <math>f</math> at <math>c</math> not existing, ''or'' both one-sided derivatives of <math>f</math> at <math>c</math> existing but being unequal.

==Related facts==

* [[Point of local extremum implies critical point for a function of multiple variables]]

==Significance==

===General comments===

The main significance of this result is that it mostly reduces an ''optimization'' problem (finding a maximum or minimum) to an ''equation-solving'' problem (finding when an expression is zero).

The reduction isn't perfect, in that it has the following caveats:

* The result is only a one-sided implication. Any point of local extremum is a critical point, but it's possible for a critical point to not be a point of local extremum. Even so, in many cases, the set of critical points is already small enough and manageable enough.
* This result is about local extrema, not absolute / global extrema. There may be many points of local extremum that aren't points of absolute extremum. Moreover, there could be many cases where an absolute extremum doesn't exist; for instance, for a function that is going to infinity as the input goes to infinity or to a point of discontunity (such as <math>x \mapsto 1/x</math> as <math>x \to 0^+</math> and <math>x \to 0^-</math>. With that being said, in many cases, we can rule out limiting behaviors of that sort, and the set of points of local extremum is small enough that it's feasible to find the absolute extremum by looking among them.

===Qualitative and existential significance===

{| class="sortable" border="1"
! Type of function for which we are interested in understanding local extrema !! Maximum number of points of local extremum !! Inference !! Example
|-
| polynomial function of degree <math>d</math> || <math>d - 1</math> || The derivative is a polynomial of degree <math>d - 1</math>, so it is defined everywhere and has at most <math>d - 1</math> real roots. Therefore, there are at most <math>d - 1</math> critical points (all of the "derivative equals zero" type, none of the "derivative is undefined" type) and therefore at most <math>d - 1</math> points of local extremum for the function. || <math>x^3 - 3x</math> of degree <math>d = 3</math> has <math>3 - 1 = 2</math> critical points: the two solutions to <math>3x^2 - 3 = 0</math>, namely <math>x = -1</math> and <math>x = 1</math>. Both of these turn out to be points of local extremum.
|-
| rational function in simplified form with numerator degree <math>d_1</math> and denominator degree <math>d_2</math> || <math>d_1 + d_2 - 1</math> || The derivative is a rational function with numerator degree at most <math>d_1 + d_2 - 1</math> and denominator a square of the original denominator. The only way for the derivative to be undefined is for the denominator to be zero, in which case the denominator of the original rational function would also be zero, so that the original rational function would also be undefined. Therefore, the derivative exists everywhere that the function is defined, and the only critical points are cases where the derivative is zero. Since the numerator is a polynomial of degree at most <math>d_1 + d_2 - 1</math>, there are therefore at most <math>d_1 + d_2 - 1</math> critical points, and therefore at most <math>d_1 + d_2 - 1</math> points of local extremum. || <math>x/(x^2 + 1)</math> with <math>d_1 = 1, d_2 = 2</math> has derivative <math>(1 - x^2)/(x^2 + 1)^2</math> and has <math>1 + 2 - 1 = 2</math> critical points, namely the solutions to <math>1 - x^2 = 0</math>, so <math>x = -1</math> and <math>x = 1</math>. Both of these turn out to be points of local extremum.
|}

===Computational feasibility significance===

As a general rule, for any family of functions <math>\mathcal{F}</math> for which we know how to find the roots of any function in the family (i.e., the points where the function is zero), we have a strategy to find critical points (and therefore points of local extremum) for any function whose ''derivative'' is in <math>\mathcal{F}</math>.

In particular, since we know how to find the roots of polynomials of degree 1 (linear polynomial) or 2 (quadratic polynomial), we can find the critical points of any degree 2 (quadratic) or degree 3 (cubic) function, as well as of any rational function with a linear numerator and quadratic denominator.

From the perspective of numerical approximation, any method for numerically approximating the ''roots'' of a function in a family <math>\mathcal{F}</math> can become a method for finding the critical points (and hence, the points of local extremum) of a function whose derivative is in <math>\mathcal{F}</math>. Some of these translations are below:

{| class="sortable" border="1"
! Root-finding method !! Corresponding optimization method
|-
| [[Newton's method for root-finding for a function of one variable]] || [[Newton's method for optimization of a function of one variable]]
|}

==Facts used==

===Statement of facts===

{| class="sortable" border="1"
! Fact no. (for reference in proof) !! Statement !! Assumption about one-sided local extremum !! Conclusion about sign of one-sided derivative !! Quick explanation in terms of the [[difference quotient]] <math>\frac{f(x) - f(c)}{x - c}</math> !! Prototypical pictures
|-
| 1 || [[uses::Local maximum from the left implies left-hand derivative is nonnegative if it exists]] (has full proof + video) || <math>f</math> has a (possibly, but not necessarily, strict) local maximum from the left at <math>c</math>. || The left-hand derivative <math>f'_-(c)</math>, if it exists, is positive or zero || In the difference quotient, the numerator is negative or zero and the denominator is negative. The quotient is thus positive or zero. Hence, so is the limit, if it exists. || [[File:Leftincreasingconcaveup.png|100px]][[File:Leftincreasingconcavedownflat.png|100px]][[File:Leftincreasingoscillatoryflat.png|100px]]
|-
| 2 || [[uses::Local minimum from the left implies left-hand derivative is nonpositive if it exists]] || <math>f</math> has a (possibly, but not necessarily, strict) local minimum from the left at <math>c</math>. || The left-hand derivative <math>f'_-(c)</math>, if it exists, is negative or zero || In the difference quotient, the numerator is positive or zero and the denominator is negative. The quotient is thus negative or zero. Hence, so is the limit, if it exists. || [[File:Leftdecreasingconcavedown.png|100px]][[File:Leftdecreasingconcaveupflat.png|100px]]
|-
| 3 || [[uses::Local maximum from the right implies right-hand derivative is nonpositive if it exists]] || <math>f</math> has a (possibly, but not necessarily, strict) local maximum from the right at <math>c</math>. || The right-hand derivative <math>f'_+(c)</math>, if it exists, is negative or zero || In the difference quotient, the numerator is negative or zero and the denominator is positive. The quotient is thus negative or zero. Hence, so is the limit, if it exists. || [[File:Rightdecreasingconcavedownnotflat.png|100px]][[File:Rightdecreasingconcaveup.png|100px]]
|-
| 4 ||[[uses::Local minimum from the right implies right-hand derivative is nonnegative if it exists]] || <math>f</math> has a (possibly, but not necessarily, strict) local minimum from the right at <math>c</math>. || The right-hand derivative <math>f'_+(c)</math>, if it exists, is positive or zero || In the difference quotient, the numerator is positive or zero and the denominator is positive. The quotient is thus positive or zero. Hence, so is the limit, if it exists. || [[File:Rightincreasingconcavedown.png|100px]][[File:Rightincreasingconcaveupnotflat.png|100px]]
|-
|}

The video below provides an intuitive explanation of the above facts. For a full proof, see the page on Fact (1).

<center>{{#widget:YouTube|id=sko6usPekeQ}}</center>

===Note on strictness===

All the facts as stated above are for not necessarily strict one-sided local extrema, i.e., we allow <math>f(x) = f(c)</math> for <math>x</math> on the immediate left or right of <math>c</math>. However, even if we impose the additional condition of strictness, we still cannot eliminate the possibility of the one-sided derivative being zero. The reason is that even though the difference quotient must now be strictly positive or strictly negative in the various cases, the one-sided derivative, which is defined as a one-sided limit of the difference quotient, may still be equal to zero. All the prototypical pictures in the previous section are pictures of strict local extrema, and some of them show a one-sided derivative of zero.

For more, see the note on strictness in the proof for Fact (1).

This observation will be crucial when we piece together the two-sided information.

===Note on sign sensitivity===

In the facts used above, we see that, when ascertaining the sign of the [[difference quotient]] <math>\frac{f(x) - f(c)}{x - c}</math>, the following are true:

* The sign of the numerator is governed by whether the point is a point of local maximum or minimum. For a point of local maximum, the numerator is negative or zero, and for a point of local minimum, the numerator is positive or zero.
* The sign of the denominator is governed by whether we are approaching from the left or the right. For a left-sided approach, the denominator is negative and for a right-sided approach, the denominator is positive.

The upshot of this is that if we change the direction of approach while preserving the nature of the local extreme value, the sign of the one-sided derivative flips. This is crucial to the proof.

==Proof==

===Proof idea===

The idea is to convert ''two-sided'' local extremum information into both its one-sided pieces, then determine the signs of the one-sided derivatives. As noted in [[#Note on sign sensitivity]], the sign conclusions for the two one-sided derivatives are opposite. We then pit these two pieces of information against each other to force the two-sided derivative, if it exists, to equal zero.

===Local maximum case===

'''Given''': A function <math>f</math>, a point <math>c</math> in the interior of the domain of <math>f</math> such that <math>f</math> attains a local maximum at <math>c</math>, i.e., <math>f(x) \le f(c)</math> for all <math>x \in (c - \delta, c + \delta)</math> for some choice of <math>\delta > 0</math>.

'''To prove''': If <math>\! f'(c)</math> exists, then <math>\!f'(c) = 0</math>

'''Proof'''

{| class="sortable" border="1"
! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation
|-
| 1 || If the left-hand derivative of <math>f</math> at <math>c</math> exists, then it is nonnegative, i.e., it is positive or zero. || Fact (1) || <math>f</math> attains a (two-sided) local maximum at <math>c</math> || || Since <math>f</math> attains a ''two-sided'' local maximum at <math>c</math>, it in particular attains a local maximum from the left. Thus, Fact (1) applies.
|-
| 2 || If the right-hand derivative of <math>f</math> at <math>c</math> exists, then it is nonpositive, i.e., it is negative or zero. || Fact (3) || <math>f</math> attains a (two-sided) local maximum at <math>c</math> || || Since <math>f</math> attains a ''two-sided'' local maximum at <math>c</math>, it in particular attains a local maximum from the right. Thus, Fact (3) applies.
|-
| 3 || If the (two-sided) derivative of <math>f</math> exists at <math>c</math>, it must be zero. || || || Steps (1), (2) || For the (two-sided) derivative to exist, ''both'' the left-hand derivative and the right-hand derivative must exist ''and'' they must be equal to each other and to the derivative. By Step (1), the derivative must be positive or zero. By Step (2), the derivative must be negative or zero. The only way both of these can be simultaneously true is if the derivative equals zero.
|}

<center>{{#widget:YouTube|id=gExgdK8xKUs}}</center>

===Local minimum case===

'''Given''': A function <math>f</math>, a point <math>c</math> in the interior of the domain of <math>f</math> such that <math>f</math> attains a local minimum at <math>c</math>, i.e., <math>f(x) \ge f(c)</math> for all <math>x \in (c - \delta, c + \delta)</math> for some choice of <math>\delta > 0</math>.

'''To prove''': If <math>f'(c)</math> exists, then <math>f'(c) = 0</math>

'''Proof'''

{| class="sortable" border="1"
! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation
|-
| 1 || If the left-hand derivative of <math>f</math> at <math>c</math> exists, then it is nonpositive, i.e., it is negative or zero. || Fact (2) || <math>f</math> attains a (two-sided) local minimum at <math>c</math> || || Since <math>f</math> attains a ''two-sided'' local minimum at <math>c</math>, it in particular attains a local minimum from the left. Thus, Fact (2) applies.
|-
| 2 || If the right-hand derivative of <math>f</math> at <math>c</math> exists, then it is nonnegative, i.e., it is positive or zero. || Fact (4) || <math>f</math> attains a (two-sided) local minimum at <math>c</math> || || Since <math>f</math> attains a ''two-sided'' local minimum at <math>c</math>, it in particular attains a local minimum from the right. Thus, Fact (4) applies.
|-
| 3 || If the (two-sided) derivative of <math>f</math> exists at <math>c</math>, it must be zero. || || || Steps (1), (2) || For the (two-sided) derivative to exist, ''both'' the left-hand derivative and the right-hand derivative must exist ''and'' they must be equal to each other and to the derivative. By Step (1), the derivative must be negative or zero. By Step (2), the derivative must be positive or zero. The only way both of these can be simultaneously true is if the derivative equals zero.
|}

===Note on strictness===

Even if we consider the case of ''strict'' two-sided local maximum or strict two-sided local minimum, we can still have either of the two types of critical point: <math>f'(c) = 0</math> or <math>f'(c)</math> not existing. The <math>f'(c) = 0</math> case continues to be possible because, for the one-sided versions, we can have the one-sided derivative equal to zero even assuming ''strict'' one-sided local extremum.

Point of local extremum implies critical point

2024-04-18T06:48:35Z

Vipul:

{{maxmin test}}
==Statement==

Suppose <math>f</math> is a [[function]] of one variable and <math>c</math> is a point in the [[interior]] of the [[domain]] of <math>f</math> (i.e., <math>f</math> is defined on an open interval containing <math>c</math>).

Suppose further that <math>c</math> is a [[point of local extremum]] for <math>f</math>, i.e., <math>f</math> attains a local extreme value (either a local maximum or a local minimum) at <math>c</math>.

Then, <math>c</math> is a [[critical point]] for <math>f</math>, i.e., either the [[derivative]] <math>f'(c)</math> equals zero or the derivative <math>f'(c)</math> does not exist.

Note that the <math>f'(c)</math> not existing case could occur in either of these ways: one or both the one-sided derivatives of <math>f</math> at <math>c</math> not existing, ''or'' both one-sided derivatives of <math>f</math> at <math>c</math> existing but being unequal.

==Related facts==

* [[Point of local extremum implies critical point for a function of multiple variables]]

==Significance==

===General comments===

The main significance of this result is that it mostly reduces an ''optimization'' problem (finding a maximum or minimum) to an ''equation-solving'' problem (finding when an expression is zero).

The reduction isn't perfect, in that it has the following caveats:

* The result is only a one-sided implication. Any point of local extremum is a critical point, but it's possible for a critical point to not be a point of local extremum. Even so, in many cases, the set of critical points is already small enough and manageable enough.
* This result is about local extrema, not absolute / global extrema. There may be many points of local extremum that aren't points of absolute extremum. Moreover, there could be many cases where an absolute extremum doesn't exist; for instance, for a function that is going to infinity as the input goes to infinity or to a point of discontunity (such as <math>x \mapsto 1/x</math> as <math>x \to 0^+</math> and <math>x \to 0^-</math>.

===Qualitative and existential significance===

{| class="sortable" border="1"
! Type of function for which we are interested in understanding local extrema !! Maximum number of points of local extremum !! Inference !! Example
|-
| polynomial function of degree <math>d</math> || <math>d - 1</math> || The derivative is a polynomial of degree <math>d - 1</math>, so it is defined everywhere and has at most <math>d - 1</math> real roots. Therefore, there are at most <math>d - 1</math> critical points (all of the "derivative equals zero" type, none of the "derivative is undefined" type) and therefore at most <math>d - 1</math> points of local extremum for the function. || <math>x^3 - 3x</math> of degree <math>d = 3</math> has <math>3 - 1 = 2</math> critical points: the two solutions to <math>3x^2 - 3 = 0</math>, namely <math>x = -1</math> and <math>x = 1</math>. Both of these turn out to be points of local extremum.
|-
| rational function in simplified form with numerator degree <math>d_1</math> and denominator degree <math>d_2</math> || <math>d_1 + d_2 - 1</math> || The derivative is a rational function with numerator degree at most <math>d_1 + d_2 - 1</math> and denominator a square of the original denominator. The only way for the derivative to be undefined is for the denominator to be zero, in which case the denominator of the original rational function would also be zero, so that the original rational function would also be undefined. Therefore, the derivative exists everywhere that the function is defined, and the only critical points are cases where the derivative is zero. Since the numerator is a polynomial of degree at most <math>d_1 + d_2 - 1</math>, there are therefore at most <math>d_1 + d_2 - 1</math> critical points, and therefore at most <math>d_1 + d_2 - 1</math> points of local extremum. || <math>x/(x^2 + 1)</math> with <math>d_1 = 1, d_2 = 2</math> has derivative <math>(1 - x^2)/(x^2 + 1)^2</math> and has <math>1 + 2 - 1 = 2</math> critical points, namely the solutions to <math>1 - x^2 = 0</math>, so <math>x = -1</math> and <math>x = 1</math>. Both of these turn out to be points of local extremum.
|}

===Computational feasibility significance===

As a general rule, for any family of functions <math>\mathcal{F}</math> for which we know how to find the roots of any function in the family (i.e., the points where the function is zero), we have a strategy to find critical points (and therefore points of local extremum) for any function whose ''derivative'' is in <math>\mathcal{F}</math>.

In particular, since we know how to find the roots of polynomials of degree 1 (linear polynomial) or 2 (quadratic polynomial), we can find the critical points of any degree 2 (quadratic) or degree 3 (cubic) function, as well as of any rational function with a linear numerator and quadratic denominator.

From the perspective of numerical approximation, any method for numerically approximating the ''roots'' of a function in a family <math>\mathcal{F}</math> can become a method for finding the critical points (and hence, the points of local extremum) of a function whose derivative is in <math>\mathcal{F}</math>. Some of these translations are below:

{| class="sortable" border="1"
! Root-finding method !! Corresponding optimization method
|-
| [[Newton's method for root-finding for a function of one variable]] || [[Newton's method for optimization of a function of one variable]]
|}

==Facts used==

===Statement of facts===

{| class="sortable" border="1"
! Fact no. (for reference in proof) !! Statement !! Assumption about one-sided local extremum !! Conclusion about sign of one-sided derivative !! Quick explanation in terms of the [[difference quotient]] <math>\frac{f(x) - f(c)}{x - c}</math> !! Prototypical pictures
|-
| 1 || [[uses::Local maximum from the left implies left-hand derivative is nonnegative if it exists]] (has full proof + video) || <math>f</math> has a (possibly, but not necessarily, strict) local maximum from the left at <math>c</math>. || The left-hand derivative <math>f'_-(c)</math>, if it exists, is positive or zero || In the difference quotient, the numerator is negative or zero and the denominator is negative. The quotient is thus positive or zero. Hence, so is the limit, if it exists. || [[File:Leftincreasingconcaveup.png|100px]][[File:Leftincreasingconcavedownflat.png|100px]][[File:Leftincreasingoscillatoryflat.png|100px]]
|-
| 2 || [[uses::Local minimum from the left implies left-hand derivative is nonpositive if it exists]] || <math>f</math> has a (possibly, but not necessarily, strict) local minimum from the left at <math>c</math>. || The left-hand derivative <math>f'_-(c)</math>, if it exists, is negative or zero || In the difference quotient, the numerator is positive or zero and the denominator is negative. The quotient is thus negative or zero. Hence, so is the limit, if it exists. || [[File:Leftdecreasingconcavedown.png|100px]][[File:Leftdecreasingconcaveupflat.png|100px]]
|-
| 3 || [[uses::Local maximum from the right implies right-hand derivative is nonpositive if it exists]] || <math>f</math> has a (possibly, but not necessarily, strict) local maximum from the right at <math>c</math>. || The right-hand derivative <math>f'_+(c)</math>, if it exists, is negative or zero || In the difference quotient, the numerator is negative or zero and the denominator is positive. The quotient is thus negative or zero. Hence, so is the limit, if it exists. || [[File:Rightdecreasingconcavedownnotflat.png|100px]][[File:Rightdecreasingconcaveup.png|100px]]
|-
| 4 ||[[uses::Local minimum from the right implies right-hand derivative is nonnegative if it exists]] || <math>f</math> has a (possibly, but not necessarily, strict) local minimum from the right at <math>c</math>. || The right-hand derivative <math>f'_+(c)</math>, if it exists, is positive or zero || In the difference quotient, the numerator is positive or zero and the denominator is positive. The quotient is thus positive or zero. Hence, so is the limit, if it exists. || [[File:Rightincreasingconcavedown.png|100px]][[File:Rightincreasingconcaveupnotflat.png|100px]]
|-
|}

The video below provides an intuitive explanation of the above facts. For a full proof, see the page on Fact (1).

<center>{{#widget:YouTube|id=sko6usPekeQ}}</center>

===Note on strictness===

All the facts as stated above are for not necessarily strict one-sided local extrema, i.e., we allow <math>f(x) = f(c)</math> for <math>x</math> on the immediate left or right of <math>c</math>. However, even if we impose the additional condition of strictness, we still cannot eliminate the possibility of the one-sided derivative being zero. The reason is that even though the difference quotient must now be strictly positive or strictly negative in the various cases, the one-sided derivative, which is defined as a one-sided limit of the difference quotient, may still be equal to zero. All the prototypical pictures in the previous section are pictures of strict local extrema, and some of them show a one-sided derivative of zero.

For more, see the note on strictness in the proof for Fact (1).

This observation will be crucial when we piece together the two-sided information.

===Note on sign sensitivity===

In the facts used above, we see that, when ascertaining the sign of the [[difference quotient]] <math>\frac{f(x) - f(c)}{x - c}</math>, the following are true:

* The sign of the numerator is governed by whether the point is a point of local maximum or minimum. For a point of local maximum, the numerator is negative or zero, and for a point of local minimum, the numerator is positive or zero.
* The sign of the denominator is governed by whether we are approaching from the left or the right. For a left-sided approach, the denominator is negative and for a right-sided approach, the denominator is positive.

The upshot of this is that if we change the direction of approach while preserving the nature of the local extreme value, the sign of the one-sided derivative flips. This is crucial to the proof.

==Proof==

===Proof idea===

The idea is to convert ''two-sided'' local extremum information into both its one-sided pieces, then determine the signs of the one-sided derivatives. As noted in [[#Note on sign sensitivity]], the sign conclusions for the two one-sided derivatives are opposite. We then pit these two pieces of information against each other to force the two-sided derivative, if it exists, to equal zero.

===Local maximum case===

'''Given''': A function <math>f</math>, a point <math>c</math> in the interior of the domain of <math>f</math> such that <math>f</math> attains a local maximum at <math>c</math>, i.e., <math>f(x) \le f(c)</math> for all <math>x \in (c - \delta, c + \delta)</math> for some choice of <math>\delta > 0</math>.

'''To prove''': If <math>\! f'(c)</math> exists, then <math>\!f'(c) = 0</math>

'''Proof'''

{| class="sortable" border="1"
! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation
|-
| 1 || If the left-hand derivative of <math>f</math> at <math>c</math> exists, then it is nonnegative, i.e., it is positive or zero. || Fact (1) || <math>f</math> attains a (two-sided) local maximum at <math>c</math> || || Since <math>f</math> attains a ''two-sided'' local maximum at <math>c</math>, it in particular attains a local maximum from the left. Thus, Fact (1) applies.
|-
| 2 || If the right-hand derivative of <math>f</math> at <math>c</math> exists, then it is nonpositive, i.e., it is negative or zero. || Fact (3) || <math>f</math> attains a (two-sided) local maximum at <math>c</math> || || Since <math>f</math> attains a ''two-sided'' local maximum at <math>c</math>, it in particular attains a local maximum from the right. Thus, Fact (3) applies.
|-
| 3 || If the (two-sided) derivative of <math>f</math> exists at <math>c</math>, it must be zero. || || || Steps (1), (2) || For the (two-sided) derivative to exist, ''both'' the left-hand derivative and the right-hand derivative must exist ''and'' they must be equal to each other and to the derivative. By Step (1), the derivative must be positive or zero. By Step (2), the derivative must be negative or zero. The only way both of these can be simultaneously true is if the derivative equals zero.
|}

<center>{{#widget:YouTube|id=gExgdK8xKUs}}</center>

===Local minimum case===

'''Given''': A function <math>f</math>, a point <math>c</math> in the interior of the domain of <math>f</math> such that <math>f</math> attains a local minimum at <math>c</math>, i.e., <math>f(x) \ge f(c)</math> for all <math>x \in (c - \delta, c + \delta)</math> for some choice of <math>\delta > 0</math>.

'''To prove''': If <math>f'(c)</math> exists, then <math>f'(c) = 0</math>

'''Proof'''

{| class="sortable" border="1"
! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation
|-
| 1 || If the left-hand derivative of <math>f</math> at <math>c</math> exists, then it is nonpositive, i.e., it is negative or zero. || Fact (2) || <math>f</math> attains a (two-sided) local minimum at <math>c</math> || || Since <math>f</math> attains a ''two-sided'' local minimum at <math>c</math>, it in particular attains a local minimum from the left. Thus, Fact (2) applies.
|-
| 2 || If the right-hand derivative of <math>f</math> at <math>c</math> exists, then it is nonnegative, i.e., it is positive or zero. || Fact (4) || <math>f</math> attains a (two-sided) local minimum at <math>c</math> || || Since <math>f</math> attains a ''two-sided'' local minimum at <math>c</math>, it in particular attains a local minimum from the right. Thus, Fact (4) applies.
|-
| 3 || If the (two-sided) derivative of <math>f</math> exists at <math>c</math>, it must be zero. || || || Steps (1), (2) || For the (two-sided) derivative to exist, ''both'' the left-hand derivative and the right-hand derivative must exist ''and'' they must be equal to each other and to the derivative. By Step (1), the derivative must be negative or zero. By Step (2), the derivative must be positive or zero. The only way both of these can be simultaneously true is if the derivative equals zero.
|}

===Note on strictness===

Even if we consider the case of ''strict'' two-sided local maximum or strict two-sided local minimum, we can still have either of the two types of critical point: <math>f'(c) = 0</math> or <math>f'(c)</math> not existing. The <math>f'(c) = 0</math> case continues to be possible because, for the one-sided versions, we can have the one-sided derivative equal to zero even assuming ''strict'' one-sided local extremum.

Point of local extremum implies critical point

2024-04-18T06:47:22Z

Vipul: /* Computational feasibility significance */

{{maxmin test}}
==Statement==

Suppose <math>f</math> is a [[function]] of one variable and <math>c</math> is a point in the [[interior]] of the [[domain]] of <math>f</math> (i.e., <math>f</math> is defined on an open interval containing <math>c</math>).

Suppose further that <math>c</math> is a [[point of local extremum]] for <math>f</math>, i.e., <math>f</math> attains a local extreme value (either a local maximum or a local minimum) at <math>c</math>.

Then, <math>c</math> is a [[critical point]] for <math>f</math>, i.e., either the [[derivative]] <math>f'(c)</math> equals zero or the derivative <math>f'(c)</math> does not exist.

Note that the <math>f'(c)</math> not existing case could occur in either of these ways: one or both the one-sided derivatives of <math>f</math> at <math>c</math> not existing, ''or'' both one-sided derivatives of <math>f</math> at <math>c</math> existing but being unequal.

==Significance==

===General comments===

The main significance of this result is that it mostly reduces an ''optimization'' problem (finding a maximum or minimum) to an ''equation-solving'' problem (finding when an expression is zero).

The reduction isn't perfect, in that it has the following caveats:

* The result is only a one-sided implication. Any point of local extremum is a critical point, but it's possible for a critical point to not be a point of local extremum. Even so, in many cases, the set of critical points is already small enough and manageable enough.
* This result is about local extrema, not absolute / global extrema. There may be many points of local extremum that aren't points of absolute extremum. Moreover, there could be many cases where an absolute extremum doesn't exist; for instance, for a function that is going to infinity as the input goes to infinity or to a point of discontunity (such as <math>x \mapsto 1/x</math> as <math>x \to 0^+</math> and <math>x \to 0^-</math>.

===Qualitative and existential significance===

{| class="sortable" border="1"
! Type of function for which we are interested in understanding local extrema !! Maximum number of points of local extremum !! Inference !! Example
|-
| polynomial function of degree <math>d</math> || <math>d - 1</math> || The derivative is a polynomial of degree <math>d - 1</math>, so it is defined everywhere and has at most <math>d - 1</math> real roots. Therefore, there are at most <math>d - 1</math> critical points (all of the "derivative equals zero" type, none of the "derivative is undefined" type) and therefore at most <math>d - 1</math> points of local extremum for the function. || <math>x^3 - 3x</math> of degree <math>d = 3</math> has <math>3 - 1 = 2</math> critical points: the two solutions to <math>3x^2 - 3 = 0</math>, namely <math>x = -1</math> and <math>x = 1</math>. Both of these turn out to be points of local extremum.
|-
| rational function in simplified form with numerator degree <math>d_1</math> and denominator degree <math>d_2</math> || <math>d_1 + d_2 - 1</math> || The derivative is a rational function with numerator degree at most <math>d_1 + d_2 - 1</math> and denominator a square of the original denominator. The only way for the derivative to be undefined is for the denominator to be zero, in which case the denominator of the original rational function would also be zero, so that the original rational function would also be undefined. Therefore, the derivative exists everywhere that the function is defined, and the only critical points are cases where the derivative is zero. Since the numerator is a polynomial of degree at most <math>d_1 + d_2 - 1</math>, there are therefore at most <math>d_1 + d_2 - 1</math> critical points, and therefore at most <math>d_1 + d_2 - 1</math> points of local extremum. || <math>x/(x^2 + 1)</math> with <math>d_1 = 1, d_2 = 2</math> has derivative <math>(1 - x^2)/(x^2 + 1)^2</math> and has <math>1 + 2 - 1 = 2</math> critical points, namely the solutions to <math>1 - x^2 = 0</math>, so <math>x = -1</math> and <math>x = 1</math>. Both of these turn out to be points of local extremum.
|}

===Computational feasibility significance===

As a general rule, for any family of functions <math>\mathcal{F}</math> for which we know how to find the roots of any function in the family (i.e., the points where the function is zero), we have a strategy to find critical points (and therefore points of local extremum) for any function whose ''derivative'' is in <math>\mathcal{F}</math>.

In particular, since we know how to find the roots of polynomials of degree 1 (linear polynomial) or 2 (quadratic polynomial), we can find the critical points of any degree 2 (quadratic) or degree 3 (cubic) function, as well as of any rational function with a linear numerator and quadratic denominator.

From the perspective of numerical approximation, any method for numerically approximating the ''roots'' of a function in a family <math>\mathcal{F}</math> can become a method for finding the critical points (and hence, the points of local extremum) of a function whose derivative is in <math>\mathcal{F}</math>. Some of these translations are below:

{| class="sortable" border="1"
! Root-finding method !! Corresponding optimization method
|-
| [[Newton's method for root-finding for a function of one variable]] || [[Newton's method for optimization of a function of one variable]]
|}

==Facts used==

===Statement of facts===

{| class="sortable" border="1"
! Fact no. (for reference in proof) !! Statement !! Assumption about one-sided local extremum !! Conclusion about sign of one-sided derivative !! Quick explanation in terms of the [[difference quotient]] <math>\frac{f(x) - f(c)}{x - c}</math> !! Prototypical pictures
|-
| 1 || [[uses::Local maximum from the left implies left-hand derivative is nonnegative if it exists]] (has full proof + video) || <math>f</math> has a (possibly, but not necessarily, strict) local maximum from the left at <math>c</math>. || The left-hand derivative <math>f'_-(c)</math>, if it exists, is positive or zero || In the difference quotient, the numerator is negative or zero and the denominator is negative. The quotient is thus positive or zero. Hence, so is the limit, if it exists. || [[File:Leftincreasingconcaveup.png|100px]][[File:Leftincreasingconcavedownflat.png|100px]][[File:Leftincreasingoscillatoryflat.png|100px]]
|-
| 2 || [[uses::Local minimum from the left implies left-hand derivative is nonpositive if it exists]] || <math>f</math> has a (possibly, but not necessarily, strict) local minimum from the left at <math>c</math>. || The left-hand derivative <math>f'_-(c)</math>, if it exists, is negative or zero || In the difference quotient, the numerator is positive or zero and the denominator is negative. The quotient is thus negative or zero. Hence, so is the limit, if it exists. || [[File:Leftdecreasingconcavedown.png|100px]][[File:Leftdecreasingconcaveupflat.png|100px]]
|-
| 3 || [[uses::Local maximum from the right implies right-hand derivative is nonpositive if it exists]] || <math>f</math> has a (possibly, but not necessarily, strict) local maximum from the right at <math>c</math>. || The right-hand derivative <math>f'_+(c)</math>, if it exists, is negative or zero || In the difference quotient, the numerator is negative or zero and the denominator is positive. The quotient is thus negative or zero. Hence, so is the limit, if it exists. || [[File:Rightdecreasingconcavedownnotflat.png|100px]][[File:Rightdecreasingconcaveup.png|100px]]
|-
| 4 ||[[uses::Local minimum from the right implies right-hand derivative is nonnegative if it exists]] || <math>f</math> has a (possibly, but not necessarily, strict) local minimum from the right at <math>c</math>. || The right-hand derivative <math>f'_+(c)</math>, if it exists, is positive or zero || In the difference quotient, the numerator is positive or zero and the denominator is positive. The quotient is thus positive or zero. Hence, so is the limit, if it exists. || [[File:Rightincreasingconcavedown.png|100px]][[File:Rightincreasingconcaveupnotflat.png|100px]]
|-
|}

The video below provides an intuitive explanation of the above facts. For a full proof, see the page on Fact (1).

<center>{{#widget:YouTube|id=sko6usPekeQ}}</center>

===Note on strictness===

All the facts as stated above are for not necessarily strict one-sided local extrema, i.e., we allow <math>f(x) = f(c)</math> for <math>x</math> on the immediate left or right of <math>c</math>. However, even if we impose the additional condition of strictness, we still cannot eliminate the possibility of the one-sided derivative being zero. The reason is that even though the difference quotient must now be strictly positive or strictly negative in the various cases, the one-sided derivative, which is defined as a one-sided limit of the difference quotient, may still be equal to zero. All the prototypical pictures in the previous section are pictures of strict local extrema, and some of them show a one-sided derivative of zero.

For more, see the note on strictness in the proof for Fact (1).

This observation will be crucial when we piece together the two-sided information.

===Note on sign sensitivity===

In the facts used above, we see that, when ascertaining the sign of the [[difference quotient]] <math>\frac{f(x) - f(c)}{x - c}</math>, the following are true:

* The sign of the numerator is governed by whether the point is a point of local maximum or minimum. For a point of local maximum, the numerator is negative or zero, and for a point of local minimum, the numerator is positive or zero.
* The sign of the denominator is governed by whether we are approaching from the left or the right. For a left-sided approach, the denominator is negative and for a right-sided approach, the denominator is positive.

The upshot of this is that if we change the direction of approach while preserving the nature of the local extreme value, the sign of the one-sided derivative flips. This is crucial to the proof.

==Proof==

===Proof idea===

The idea is to convert ''two-sided'' local extremum information into both its one-sided pieces, then determine the signs of the one-sided derivatives. As noted in [[#Note on sign sensitivity]], the sign conclusions for the two one-sided derivatives are opposite. We then pit these two pieces of information against each other to force the two-sided derivative, if it exists, to equal zero.

===Local maximum case===

'''Given''': A function <math>f</math>, a point <math>c</math> in the interior of the domain of <math>f</math> such that <math>f</math> attains a local maximum at <math>c</math>, i.e., <math>f(x) \le f(c)</math> for all <math>x \in (c - \delta, c + \delta)</math> for some choice of <math>\delta > 0</math>.

'''To prove''': If <math>\! f'(c)</math> exists, then <math>\!f'(c) = 0</math>

'''Proof'''

{| class="sortable" border="1"
! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation
|-
| 1 || If the left-hand derivative of <math>f</math> at <math>c</math> exists, then it is nonnegative, i.e., it is positive or zero. || Fact (1) || <math>f</math> attains a (two-sided) local maximum at <math>c</math> || || Since <math>f</math> attains a ''two-sided'' local maximum at <math>c</math>, it in particular attains a local maximum from the left. Thus, Fact (1) applies.
|-
| 2 || If the right-hand derivative of <math>f</math> at <math>c</math> exists, then it is nonpositive, i.e., it is negative or zero. || Fact (3) || <math>f</math> attains a (two-sided) local maximum at <math>c</math> || || Since <math>f</math> attains a ''two-sided'' local maximum at <math>c</math>, it in particular attains a local maximum from the right. Thus, Fact (3) applies.
|-
| 3 || If the (two-sided) derivative of <math>f</math> exists at <math>c</math>, it must be zero. || || || Steps (1), (2) || For the (two-sided) derivative to exist, ''both'' the left-hand derivative and the right-hand derivative must exist ''and'' they must be equal to each other and to the derivative. By Step (1), the derivative must be positive or zero. By Step (2), the derivative must be negative or zero. The only way both of these can be simultaneously true is if the derivative equals zero.
|}

<center>{{#widget:YouTube|id=gExgdK8xKUs}}</center>

===Local minimum case===

'''Given''': A function <math>f</math>, a point <math>c</math> in the interior of the domain of <math>f</math> such that <math>f</math> attains a local minimum at <math>c</math>, i.e., <math>f(x) \ge f(c)</math> for all <math>x \in (c - \delta, c + \delta)</math> for some choice of <math>\delta > 0</math>.

'''To prove''': If <math>f'(c)</math> exists, then <math>f'(c) = 0</math>

'''Proof'''

{| class="sortable" border="1"
! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation
|-
| 1 || If the left-hand derivative of <math>f</math> at <math>c</math> exists, then it is nonpositive, i.e., it is negative or zero. || Fact (2) || <math>f</math> attains a (two-sided) local minimum at <math>c</math> || || Since <math>f</math> attains a ''two-sided'' local minimum at <math>c</math>, it in particular attains a local minimum from the left. Thus, Fact (2) applies.
|-
| 2 || If the right-hand derivative of <math>f</math> at <math>c</math> exists, then it is nonnegative, i.e., it is positive or zero. || Fact (4) || <math>f</math> attains a (two-sided) local minimum at <math>c</math> || || Since <math>f</math> attains a ''two-sided'' local minimum at <math>c</math>, it in particular attains a local minimum from the right. Thus, Fact (4) applies.
|-
| 3 || If the (two-sided) derivative of <math>f</math> exists at <math>c</math>, it must be zero. || || || Steps (1), (2) || For the (two-sided) derivative to exist, ''both'' the left-hand derivative and the right-hand derivative must exist ''and'' they must be equal to each other and to the derivative. By Step (1), the derivative must be negative or zero. By Step (2), the derivative must be positive or zero. The only way both of these can be simultaneously true is if the derivative equals zero.
|}

===Note on strictness===

Even if we consider the case of ''strict'' two-sided local maximum or strict two-sided local minimum, we can still have either of the two types of critical point: <math>f'(c) = 0</math> or <math>f'(c)</math> not existing. The <math>f'(c) = 0</math> case continues to be possible because, for the one-sided versions, we can have the one-sided derivative equal to zero even assuming ''strict'' one-sided local extremum.

Point of local extremum implies critical point

2024-04-18T06:46:46Z

Vipul: /* Significance */

{{maxmin test}}
==Statement==

Suppose <math>f</math> is a [[function]] of one variable and <math>c</math> is a point in the [[interior]] of the [[domain]] of <math>f</math> (i.e., <math>f</math> is defined on an open interval containing <math>c</math>).

Suppose further that <math>c</math> is a [[point of local extremum]] for <math>f</math>, i.e., <math>f</math> attains a local extreme value (either a local maximum or a local minimum) at <math>c</math>.

Then, <math>c</math> is a [[critical point]] for <math>f</math>, i.e., either the [[derivative]] <math>f'(c)</math> equals zero or the derivative <math>f'(c)</math> does not exist.

Note that the <math>f'(c)</math> not existing case could occur in either of these ways: one or both the one-sided derivatives of <math>f</math> at <math>c</math> not existing, ''or'' both one-sided derivatives of <math>f</math> at <math>c</math> existing but being unequal.

==Significance==

===General comments===

The main significance of this result is that it mostly reduces an ''optimization'' problem (finding a maximum or minimum) to an ''equation-solving'' problem (finding when an expression is zero).

The reduction isn't perfect, in that it has the following caveats:

* The result is only a one-sided implication. Any point of local extremum is a critical point, but it's possible for a critical point to not be a point of local extremum. Even so, in many cases, the set of critical points is already small enough and manageable enough.
* This result is about local extrema, not absolute / global extrema. There may be many points of local extremum that aren't points of absolute extremum. Moreover, there could be many cases where an absolute extremum doesn't exist; for instance, for a function that is going to infinity as the input goes to infinity or to a point of discontunity (such as <math>x \mapsto 1/x</math> as <math>x \to 0^+</math> and <math>x \to 0^-</math>.

===Qualitative and existential significance===

{| class="sortable" border="1"
! Type of function for which we are interested in understanding local extrema !! Maximum number of points of local extremum !! Inference !! Example
|-
| polynomial function of degree <math>d</math> || <math>d - 1</math> || The derivative is a polynomial of degree <math>d - 1</math>, so it is defined everywhere and has at most <math>d - 1</math> real roots. Therefore, there are at most <math>d - 1</math> critical points (all of the "derivative equals zero" type, none of the "derivative is undefined" type) and therefore at most <math>d - 1</math> points of local extremum for the function. || <math>x^3 - 3x</math> of degree <math>d = 3</math> has <math>3 - 1 = 2</math> critical points: the two solutions to <math>3x^2 - 3 = 0</math>, namely <math>x = -1</math> and <math>x = 1</math>. Both of these turn out to be points of local extremum.
|-
| rational function in simplified form with numerator degree <math>d_1</math> and denominator degree <math>d_2</math> || <math>d_1 + d_2 - 1</math> || The derivative is a rational function with numerator degree at most <math>d_1 + d_2 - 1</math> and denominator a square of the original denominator. The only way for the derivative to be undefined is for the denominator to be zero, in which case the denominator of the original rational function would also be zero, so that the original rational function would also be undefined. Therefore, the derivative exists everywhere that the function is defined, and the only critical points are cases where the derivative is zero. Since the numerator is a polynomial of degree at most <math>d_1 + d_2 - 1</math>, there are therefore at most <math>d_1 + d_2 - 1</math> critical points, and therefore at most <math>d_1 + d_2 - 1</math> points of local extremum. || <math>x/(x^2 + 1)</math> with <math>d_1 = 1, d_2 = 2</math> has derivative <math>(1 - x^2)/(x^2 + 1)^2</math> and has <math>1 + 2 - 1 = 2</math> critical points, namely the solutions to <math>1 - x^2 = 0</math>, so <math>x = -1</math> and <math>x = 1</math>. Both of these turn out to be points of local extremum.
|}

===Computational feasibility significance===

As a general rule, for any family of functions <math>\mathcal{F}</math> for which we know how to find the roots of any function in the family (i.e., the points where the function is zero), we have a strategy to find critical points (and therefore points of local extremum) for any function whose ''derivative'' is in <math>\mathcal{F}</math>.

In particular, since we know how to find the roots of polynomials of degree 1 (linear polynomial) or 2 (quadratic polynomial), we can find the critical points of any degree 2 (quadratic) or degree 3 (cubic) function, as well as of any rational function with a linear numerator and quadratic denominator.

From the perspective of numerical approximation, any method for numerically approximating the ''roots'' of a function in a family <math>\mathcal{F}</math> can become a method for finding the critical points (and hence, the points of local extremum) of a function whose derivative is in <math>\mathcal{F}</math>. Some of these translations are below:

{| class="sortable" border="1"
! Root-finding method !! Optimization method
|-
| [[Newton's method for root-finding for a function of one variable]] || [[Newton's method for optimization of a function of one variable]]
|}

==Facts used==

===Statement of facts===

{| class="sortable" border="1"
! Fact no. (for reference in proof) !! Statement !! Assumption about one-sided local extremum !! Conclusion about sign of one-sided derivative !! Quick explanation in terms of the [[difference quotient]] <math>\frac{f(x) - f(c)}{x - c}</math> !! Prototypical pictures
|-
| 1 || [[uses::Local maximum from the left implies left-hand derivative is nonnegative if it exists]] (has full proof + video) || <math>f</math> has a (possibly, but not necessarily, strict) local maximum from the left at <math>c</math>. || The left-hand derivative <math>f'_-(c)</math>, if it exists, is positive or zero || In the difference quotient, the numerator is negative or zero and the denominator is negative. The quotient is thus positive or zero. Hence, so is the limit, if it exists. || [[File:Leftincreasingconcaveup.png|100px]][[File:Leftincreasingconcavedownflat.png|100px]][[File:Leftincreasingoscillatoryflat.png|100px]]
|-
| 2 || [[uses::Local minimum from the left implies left-hand derivative is nonpositive if it exists]] || <math>f</math> has a (possibly, but not necessarily, strict) local minimum from the left at <math>c</math>. || The left-hand derivative <math>f'_-(c)</math>, if it exists, is negative or zero || In the difference quotient, the numerator is positive or zero and the denominator is negative. The quotient is thus negative or zero. Hence, so is the limit, if it exists. || [[File:Leftdecreasingconcavedown.png|100px]][[File:Leftdecreasingconcaveupflat.png|100px]]
|-
| 3 || [[uses::Local maximum from the right implies right-hand derivative is nonpositive if it exists]] || <math>f</math> has a (possibly, but not necessarily, strict) local maximum from the right at <math>c</math>. || The right-hand derivative <math>f'_+(c)</math>, if it exists, is negative or zero || In the difference quotient, the numerator is negative or zero and the denominator is positive. The quotient is thus negative or zero. Hence, so is the limit, if it exists. || [[File:Rightdecreasingconcavedownnotflat.png|100px]][[File:Rightdecreasingconcaveup.png|100px]]
|-
| 4 ||[[uses::Local minimum from the right implies right-hand derivative is nonnegative if it exists]] || <math>f</math> has a (possibly, but not necessarily, strict) local minimum from the right at <math>c</math>. || The right-hand derivative <math>f'_+(c)</math>, if it exists, is positive or zero || In the difference quotient, the numerator is positive or zero and the denominator is positive. The quotient is thus positive or zero. Hence, so is the limit, if it exists. || [[File:Rightincreasingconcavedown.png|100px]][[File:Rightincreasingconcaveupnotflat.png|100px]]
|-
|}

The video below provides an intuitive explanation of the above facts. For a full proof, see the page on Fact (1).

<center>{{#widget:YouTube|id=sko6usPekeQ}}</center>

===Note on strictness===

All the facts as stated above are for not necessarily strict one-sided local extrema, i.e., we allow <math>f(x) = f(c)</math> for <math>x</math> on the immediate left or right of <math>c</math>. However, even if we impose the additional condition of strictness, we still cannot eliminate the possibility of the one-sided derivative being zero. The reason is that even though the difference quotient must now be strictly positive or strictly negative in the various cases, the one-sided derivative, which is defined as a one-sided limit of the difference quotient, may still be equal to zero. All the prototypical pictures in the previous section are pictures of strict local extrema, and some of them show a one-sided derivative of zero.

For more, see the note on strictness in the proof for Fact (1).

This observation will be crucial when we piece together the two-sided information.

===Note on sign sensitivity===

In the facts used above, we see that, when ascertaining the sign of the [[difference quotient]] <math>\frac{f(x) - f(c)}{x - c}</math>, the following are true:

* The sign of the numerator is governed by whether the point is a point of local maximum or minimum. For a point of local maximum, the numerator is negative or zero, and for a point of local minimum, the numerator is positive or zero.
* The sign of the denominator is governed by whether we are approaching from the left or the right. For a left-sided approach, the denominator is negative and for a right-sided approach, the denominator is positive.

The upshot of this is that if we change the direction of approach while preserving the nature of the local extreme value, the sign of the one-sided derivative flips. This is crucial to the proof.

==Proof==

===Proof idea===

The idea is to convert ''two-sided'' local extremum information into both its one-sided pieces, then determine the signs of the one-sided derivatives. As noted in [[#Note on sign sensitivity]], the sign conclusions for the two one-sided derivatives are opposite. We then pit these two pieces of information against each other to force the two-sided derivative, if it exists, to equal zero.

===Local maximum case===

'''Given''': A function <math>f</math>, a point <math>c</math> in the interior of the domain of <math>f</math> such that <math>f</math> attains a local maximum at <math>c</math>, i.e., <math>f(x) \le f(c)</math> for all <math>x \in (c - \delta, c + \delta)</math> for some choice of <math>\delta > 0</math>.

'''To prove''': If <math>\! f'(c)</math> exists, then <math>\!f'(c) = 0</math>

'''Proof'''

{| class="sortable" border="1"
! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation
|-
| 1 || If the left-hand derivative of <math>f</math> at <math>c</math> exists, then it is nonnegative, i.e., it is positive or zero. || Fact (1) || <math>f</math> attains a (two-sided) local maximum at <math>c</math> || || Since <math>f</math> attains a ''two-sided'' local maximum at <math>c</math>, it in particular attains a local maximum from the left. Thus, Fact (1) applies.
|-
| 2 || If the right-hand derivative of <math>f</math> at <math>c</math> exists, then it is nonpositive, i.e., it is negative or zero. || Fact (3) || <math>f</math> attains a (two-sided) local maximum at <math>c</math> || || Since <math>f</math> attains a ''two-sided'' local maximum at <math>c</math>, it in particular attains a local maximum from the right. Thus, Fact (3) applies.
|-
| 3 || If the (two-sided) derivative of <math>f</math> exists at <math>c</math>, it must be zero. || || || Steps (1), (2) || For the (two-sided) derivative to exist, ''both'' the left-hand derivative and the right-hand derivative must exist ''and'' they must be equal to each other and to the derivative. By Step (1), the derivative must be positive or zero. By Step (2), the derivative must be negative or zero. The only way both of these can be simultaneously true is if the derivative equals zero.
|}

<center>{{#widget:YouTube|id=gExgdK8xKUs}}</center>

===Local minimum case===

'''Given''': A function <math>f</math>, a point <math>c</math> in the interior of the domain of <math>f</math> such that <math>f</math> attains a local minimum at <math>c</math>, i.e., <math>f(x) \ge f(c)</math> for all <math>x \in (c - \delta, c + \delta)</math> for some choice of <math>\delta > 0</math>.

'''To prove''': If <math>f'(c)</math> exists, then <math>f'(c) = 0</math>

'''Proof'''

{| class="sortable" border="1"
! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation
|-
| 1 || If the left-hand derivative of <math>f</math> at <math>c</math> exists, then it is nonpositive, i.e., it is negative or zero. || Fact (2) || <math>f</math> attains a (two-sided) local minimum at <math>c</math> || || Since <math>f</math> attains a ''two-sided'' local minimum at <math>c</math>, it in particular attains a local minimum from the left. Thus, Fact (2) applies.
|-
| 2 || If the right-hand derivative of <math>f</math> at <math>c</math> exists, then it is nonnegative, i.e., it is positive or zero. || Fact (4) || <math>f</math> attains a (two-sided) local minimum at <math>c</math> || || Since <math>f</math> attains a ''two-sided'' local minimum at <math>c</math>, it in particular attains a local minimum from the right. Thus, Fact (4) applies.
|-
| 3 || If the (two-sided) derivative of <math>f</math> exists at <math>c</math>, it must be zero. || || || Steps (1), (2) || For the (two-sided) derivative to exist, ''both'' the left-hand derivative and the right-hand derivative must exist ''and'' they must be equal to each other and to the derivative. By Step (1), the derivative must be negative or zero. By Step (2), the derivative must be positive or zero. The only way both of these can be simultaneously true is if the derivative equals zero.
|}

===Note on strictness===

Even if we consider the case of ''strict'' two-sided local maximum or strict two-sided local minimum, we can still have either of the two types of critical point: <math>f'(c) = 0</math> or <math>f'(c)</math> not existing. The <math>f'(c) = 0</math> case continues to be possible because, for the one-sided versions, we can have the one-sided derivative equal to zero even assuming ''strict'' one-sided local extremum.

Point of local extremum implies critical point

2024-04-18T06:42:39Z

Vipul: /* Significance */

{{maxmin test}}
==Statement==

Suppose <math>f</math> is a [[function]] of one variable and <math>c</math> is a point in the [[interior]] of the [[domain]] of <math>f</math> (i.e., <math>f</math> is defined on an open interval containing <math>c</math>).

Suppose further that <math>c</math> is a [[point of local extremum]] for <math>f</math>, i.e., <math>f</math> attains a local extreme value (either a local maximum or a local minimum) at <math>c</math>.

Then, <math>c</math> is a [[critical point]] for <math>f</math>, i.e., either the [[derivative]] <math>f'(c)</math> equals zero or the derivative <math>f'(c)</math> does not exist.

Note that the <math>f'(c)</math> not existing case could occur in either of these ways: one or both the one-sided derivatives of <math>f</math> at <math>c</math> not existing, ''or'' both one-sided derivatives of <math>f</math> at <math>c</math> existing but being unequal.

==Significance==

===General comments===

The main significance of this result is that it mostly reduces an ''optimization'' problem (finding a maximum or minimum) to an ''equation-solving'' problem (finding when an expression is zero).

The reduction isn't perfect, in that it has the following caveats:

* The result is only a one-sided implication. Any point of local extremum is a critical point, but it's possible for a critical point to not be a point of local extremum. Even so, in many cases, the set of critical points is already small enough and manageable enough.
* This result is about local extrema, not absolute / global extrema. There may be many points of local extremum that aren't points of absolute extremum. Moreover, there could be many cases where an absolute extremum doesn't exist; for instance, for a function that is going to infinity as the input goes to infinity or to a point of discontunity (such as <math>x \mapsto 1/x</math> as <math>x \to 0^+</math> and <math>x \to 0^-</math>.

===Qualitative and existential significance===

{| class="sortable" border="1"
! Type of function for which we are interested in understanding local extrema !! Maximum number of points of local extremum !! Inference !! Example
|-
| polynomial function of degree <math>d</math> || <math>d - 1</math> || The derivative is a polynomial of degree <math>d - 1</math>, so it is defined everywhere and has at most <math>d - 1</math> real roots. Therefore, there are at most <math>d - 1</math> critical points (all of the "derivative equals zero" type, none of the "derivative is undefined" type) and therefore at most <math>d - 1</math> points of local extremum for the function. || <math>x^3 - 3x</math> of degree <math>d = 3</math> has <math>3 - 1 = 2</math> critical points: the two solutions to <math>3x^2 - 3 = 0</math>, namely <math>x = -1</math> and <math>x = 1</math>. Both of these turn out to be points of local extremum.
|-
| rational function in simplified form with numerator degree <math>d_1</math> and denominator degree <math>d_2</math> || <math>d_1 + d_2 - 1</math> || The derivative is a rational function with numerator degree at most <math>d_1 + d_2 - 1</math> and denominator a square of the original denominator. The only way for the derivative to be undefined is for the denominator to be zero, in which case the denominator of the original rational function would also be zero, so that the original rational function would also be undefined. Therefore, the derivative exists everywhere that the function is defined, and the only critical points are cases where the derivative is zero. Since the numerator is a polynomial of degree at most <math>d_1 + d_2 - 1</math>, there are therefore at most <math>d_1 + d_2 - 1</math> critical points, and therefore at most <math>d_1 + d_2 - 1</math> points of local extremum. || <math>x/(x^2 + 1)</math> with <math>d_1 = 1, d_2 = 2</math> has derivative <math>(1 - x^2)/(x^2 + 1)^2</math> and has <math>1 + 2 - 1 = 2</math> critical points, namely the solutions to <math>1 - x^2 = 0</math>, so <math>x = -1</math> and <math>x = 1</math>. Both of these turn out to be points of local extremum.
|}

===Computational feasibility significance===

As a general rule, for any family of functions <math>\mathcal{F}</math> for which we know how to find the roots of any function in the family (i.e., the points where the function is zero), we have a strategy to find critical points (and therefore points of local extremum) for any function whose ''derivative'' is in <math>\mathcal{F}</math>.

In particular, since we know how to find the roots of polynomials of degree 1 (linear polynomial) or 2 (quadratic polynomial), we can find the critical points of any degree 2 (quadratic) or degree 3 (cubic) function, as well as of any rational function with a linear numerator and quadratic denominator.

==Facts used==

===Statement of facts===

{| class="sortable" border="1"
! Fact no. (for reference in proof) !! Statement !! Assumption about one-sided local extremum !! Conclusion about sign of one-sided derivative !! Quick explanation in terms of the [[difference quotient]] <math>\frac{f(x) - f(c)}{x - c}</math> !! Prototypical pictures
|-
| 1 || [[uses::Local maximum from the left implies left-hand derivative is nonnegative if it exists]] (has full proof + video) || <math>f</math> has a (possibly, but not necessarily, strict) local maximum from the left at <math>c</math>. || The left-hand derivative <math>f'_-(c)</math>, if it exists, is positive or zero || In the difference quotient, the numerator is negative or zero and the denominator is negative. The quotient is thus positive or zero. Hence, so is the limit, if it exists. || [[File:Leftincreasingconcaveup.png|100px]][[File:Leftincreasingconcavedownflat.png|100px]][[File:Leftincreasingoscillatoryflat.png|100px]]
|-
| 2 || [[uses::Local minimum from the left implies left-hand derivative is nonpositive if it exists]] || <math>f</math> has a (possibly, but not necessarily, strict) local minimum from the left at <math>c</math>. || The left-hand derivative <math>f'_-(c)</math>, if it exists, is negative or zero || In the difference quotient, the numerator is positive or zero and the denominator is negative. The quotient is thus negative or zero. Hence, so is the limit, if it exists. || [[File:Leftdecreasingconcavedown.png|100px]][[File:Leftdecreasingconcaveupflat.png|100px]]
|-
| 3 || [[uses::Local maximum from the right implies right-hand derivative is nonpositive if it exists]] || <math>f</math> has a (possibly, but not necessarily, strict) local maximum from the right at <math>c</math>. || The right-hand derivative <math>f'_+(c)</math>, if it exists, is negative or zero || In the difference quotient, the numerator is negative or zero and the denominator is positive. The quotient is thus negative or zero. Hence, so is the limit, if it exists. || [[File:Rightdecreasingconcavedownnotflat.png|100px]][[File:Rightdecreasingconcaveup.png|100px]]
|-
| 4 ||[[uses::Local minimum from the right implies right-hand derivative is nonnegative if it exists]] || <math>f</math> has a (possibly, but not necessarily, strict) local minimum from the right at <math>c</math>. || The right-hand derivative <math>f'_+(c)</math>, if it exists, is positive or zero || In the difference quotient, the numerator is positive or zero and the denominator is positive. The quotient is thus positive or zero. Hence, so is the limit, if it exists. || [[File:Rightincreasingconcavedown.png|100px]][[File:Rightincreasingconcaveupnotflat.png|100px]]
|-
|}

The video below provides an intuitive explanation of the above facts. For a full proof, see the page on Fact (1).

<center>{{#widget:YouTube|id=sko6usPekeQ}}</center>

===Note on strictness===

All the facts as stated above are for not necessarily strict one-sided local extrema, i.e., we allow <math>f(x) = f(c)</math> for <math>x</math> on the immediate left or right of <math>c</math>. However, even if we impose the additional condition of strictness, we still cannot eliminate the possibility of the one-sided derivative being zero. The reason is that even though the difference quotient must now be strictly positive or strictly negative in the various cases, the one-sided derivative, which is defined as a one-sided limit of the difference quotient, may still be equal to zero. All the prototypical pictures in the previous section are pictures of strict local extrema, and some of them show a one-sided derivative of zero.

For more, see the note on strictness in the proof for Fact (1).

This observation will be crucial when we piece together the two-sided information.

===Note on sign sensitivity===

In the facts used above, we see that, when ascertaining the sign of the [[difference quotient]] <math>\frac{f(x) - f(c)}{x - c}</math>, the following are true:

* The sign of the numerator is governed by whether the point is a point of local maximum or minimum. For a point of local maximum, the numerator is negative or zero, and for a point of local minimum, the numerator is positive or zero.
* The sign of the denominator is governed by whether we are approaching from the left or the right. For a left-sided approach, the denominator is negative and for a right-sided approach, the denominator is positive.

The upshot of this is that if we change the direction of approach while preserving the nature of the local extreme value, the sign of the one-sided derivative flips. This is crucial to the proof.

==Proof==

===Proof idea===

The idea is to convert ''two-sided'' local extremum information into both its one-sided pieces, then determine the signs of the one-sided derivatives. As noted in [[#Note on sign sensitivity]], the sign conclusions for the two one-sided derivatives are opposite. We then pit these two pieces of information against each other to force the two-sided derivative, if it exists, to equal zero.

===Local maximum case===

'''Given''': A function <math>f</math>, a point <math>c</math> in the interior of the domain of <math>f</math> such that <math>f</math> attains a local maximum at <math>c</math>, i.e., <math>f(x) \le f(c)</math> for all <math>x \in (c - \delta, c + \delta)</math> for some choice of <math>\delta > 0</math>.

'''To prove''': If <math>\! f'(c)</math> exists, then <math>\!f'(c) = 0</math>

'''Proof'''

{| class="sortable" border="1"
! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation
|-
| 1 || If the left-hand derivative of <math>f</math> at <math>c</math> exists, then it is nonnegative, i.e., it is positive or zero. || Fact (1) || <math>f</math> attains a (two-sided) local maximum at <math>c</math> || || Since <math>f</math> attains a ''two-sided'' local maximum at <math>c</math>, it in particular attains a local maximum from the left. Thus, Fact (1) applies.
|-
| 2 || If the right-hand derivative of <math>f</math> at <math>c</math> exists, then it is nonpositive, i.e., it is negative or zero. || Fact (3) || <math>f</math> attains a (two-sided) local maximum at <math>c</math> || || Since <math>f</math> attains a ''two-sided'' local maximum at <math>c</math>, it in particular attains a local maximum from the right. Thus, Fact (3) applies.
|-
| 3 || If the (two-sided) derivative of <math>f</math> exists at <math>c</math>, it must be zero. || || || Steps (1), (2) || For the (two-sided) derivative to exist, ''both'' the left-hand derivative and the right-hand derivative must exist ''and'' they must be equal to each other and to the derivative. By Step (1), the derivative must be positive or zero. By Step (2), the derivative must be negative or zero. The only way both of these can be simultaneously true is if the derivative equals zero.
|}

<center>{{#widget:YouTube|id=gExgdK8xKUs}}</center>

===Local minimum case===

'''Given''': A function <math>f</math>, a point <math>c</math> in the interior of the domain of <math>f</math> such that <math>f</math> attains a local minimum at <math>c</math>, i.e., <math>f(x) \ge f(c)</math> for all <math>x \in (c - \delta, c + \delta)</math> for some choice of <math>\delta > 0</math>.

'''To prove''': If <math>f'(c)</math> exists, then <math>f'(c) = 0</math>

'''Proof'''

{| class="sortable" border="1"
! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation
|-
| 1 || If the left-hand derivative of <math>f</math> at <math>c</math> exists, then it is nonpositive, i.e., it is negative or zero. || Fact (2) || <math>f</math> attains a (two-sided) local minimum at <math>c</math> || || Since <math>f</math> attains a ''two-sided'' local minimum at <math>c</math>, it in particular attains a local minimum from the left. Thus, Fact (2) applies.
|-
| 2 || If the right-hand derivative of <math>f</math> at <math>c</math> exists, then it is nonnegative, i.e., it is positive or zero. || Fact (4) || <math>f</math> attains a (two-sided) local minimum at <math>c</math> || || Since <math>f</math> attains a ''two-sided'' local minimum at <math>c</math>, it in particular attains a local minimum from the right. Thus, Fact (4) applies.
|-
| 3 || If the (two-sided) derivative of <math>f</math> exists at <math>c</math>, it must be zero. || || || Steps (1), (2) || For the (two-sided) derivative to exist, ''both'' the left-hand derivative and the right-hand derivative must exist ''and'' they must be equal to each other and to the derivative. By Step (1), the derivative must be negative or zero. By Step (2), the derivative must be positive or zero. The only way both of these can be simultaneously true is if the derivative equals zero.
|}

===Note on strictness===

Even if we consider the case of ''strict'' two-sided local maximum or strict two-sided local minimum, we can still have either of the two types of critical point: <math>f'(c) = 0</math> or <math>f'(c)</math> not existing. The <math>f'(c) = 0</math> case continues to be possible because, for the one-sided versions, we can have the one-sided derivative equal to zero even assuming ''strict'' one-sided local extremum.

Point of local extremum implies critical point

2024-04-18T06:38:32Z

Vipul: /* Qualitative and existential significance */

{{maxmin test}}
==Statement==

Suppose <math>f</math> is a [[function]] of one variable and <math>c</math> is a point in the [[interior]] of the [[domain]] of <math>f</math> (i.e., <math>f</math> is defined on an open interval containing <math>c</math>).

Suppose further that <math>c</math> is a [[point of local extremum]] for <math>f</math>, i.e., <math>f</math> attains a local extreme value (either a local maximum or a local minimum) at <math>c</math>.

Then, <math>c</math> is a [[critical point]] for <math>f</math>, i.e., either the [[derivative]] <math>f'(c)</math> equals zero or the derivative <math>f'(c)</math> does not exist.

Note that the <math>f'(c)</math> not existing case could occur in either of these ways: one or both the one-sided derivatives of <math>f</math> at <math>c</math> not existing, ''or'' both one-sided derivatives of <math>f</math> at <math>c</math> existing but being unequal.

==Significance==

===General comments===

The main significance of this result is that it mostly reduces an ''optimization'' problem (finding a maximum or minimum) to an ''equation-solving'' problem (finding when an expression is zero).

The reduction isn't perfect, in that it has the following caveats:

* The result is only a one-sided implication. Any point of local extremum is a critical point, but it's possible for a critical point to not be a point of local extremum. Even so, in many cases, the set of critical points is already small enough and manageable enough.
* This result is about local extrema, not absolute / global extrema. There may be many points of local extremum that aren't points of absolute extremum. Moreover, there could be many cases where an absolute extremum doesn't exist; for instance, for a function that is going to infinity as the input goes to infinity or to a point of discontunity (such as <math>x \mapsto 1/x</math> as <math>x \to 0^+</math> and <math>x \to 0^-</math>.

===Qualitative and existential significance===

{| class="sortable" border="1"
! Type of function for which we are interested in understanding local extrema !! Maximum number of points of local extremum !! Inference !! Example
|-
| polynomial function of degree <math>d</math> || <math>d - 1</math> || The derivative is a polynomial of degree <math>d - 1</math>, so it is defined everywhere and has at most <math>d - 1</math> real roots. Therefore, there are at most <math>d - 1</math> critical points (all of the "derivative equals zero" type, none of the "derivative is undefined" type) and therefore at most <math>d - 1</math> points of local extremum for the function. || <math>x^3 - 3x</math> of degree <math>d = 3</math> has <math>3 - 1 = 2</math> critical points: the two solutions to <math>3x^2 - 3 = 0</math>, namely <math>x = -1</math> and <math>x = 1</math>. Both of these turn out to be points of local extremum.
|-
| rational function in simplified form with numerator degree <math>d_1</math> and denominator degree <math>d_2</math> || <math>d_1 + d_2 - 1</math> || The derivative is a rational function with numerator degree at most <math>d_1 + d_2 - 1</math> and denominator a square of the original denominator. The only way for the derivative to be undefined is for the denominator to be zero, in which case the denominator of the original rational function would also be zero, so that the original rational function would also be undefined. Therefore, the derivative exists everywhere that the function is defined, and the only critical points are cases where the derivative is zero. Since the numerator is a polynomial of degree at most <math>d_1 + d_2 - 1</math>, there are therefore at most <math>d_1 + d_2 - 1</math> critical points, and therefore at most <math>d_1 + d_2 - 1</math> points of local extremum. || <math>x/(x^2 + 1)</math> with <math>d_1 = 1, d_2 = 2</math> has derivative <math>(1 - x^2)/(x^2 + 1)^2</math> and has <math>1 + 2 - 1 = 2</math> critical points, namely the solutions to <math>1 - x^2 = 0</math>, so <math>x = -1</math> and <math>x = 1</math>. Both of these turn out to be points of local extremum.
|}

==Facts used==

===Statement of facts===

{| class="sortable" border="1"
! Fact no. (for reference in proof) !! Statement !! Assumption about one-sided local extremum !! Conclusion about sign of one-sided derivative !! Quick explanation in terms of the [[difference quotient]] <math>\frac{f(x) - f(c)}{x - c}</math> !! Prototypical pictures
|-
| 1 || [[uses::Local maximum from the left implies left-hand derivative is nonnegative if it exists]] (has full proof + video) || <math>f</math> has a (possibly, but not necessarily, strict) local maximum from the left at <math>c</math>. || The left-hand derivative <math>f'_-(c)</math>, if it exists, is positive or zero || In the difference quotient, the numerator is negative or zero and the denominator is negative. The quotient is thus positive or zero. Hence, so is the limit, if it exists. || [[File:Leftincreasingconcaveup.png|100px]][[File:Leftincreasingconcavedownflat.png|100px]][[File:Leftincreasingoscillatoryflat.png|100px]]
|-
| 2 || [[uses::Local minimum from the left implies left-hand derivative is nonpositive if it exists]] || <math>f</math> has a (possibly, but not necessarily, strict) local minimum from the left at <math>c</math>. || The left-hand derivative <math>f'_-(c)</math>, if it exists, is negative or zero || In the difference quotient, the numerator is positive or zero and the denominator is negative. The quotient is thus negative or zero. Hence, so is the limit, if it exists. || [[File:Leftdecreasingconcavedown.png|100px]][[File:Leftdecreasingconcaveupflat.png|100px]]
|-
| 3 || [[uses::Local maximum from the right implies right-hand derivative is nonpositive if it exists]] || <math>f</math> has a (possibly, but not necessarily, strict) local maximum from the right at <math>c</math>. || The right-hand derivative <math>f'_+(c)</math>, if it exists, is negative or zero || In the difference quotient, the numerator is negative or zero and the denominator is positive. The quotient is thus negative or zero. Hence, so is the limit, if it exists. || [[File:Rightdecreasingconcavedownnotflat.png|100px]][[File:Rightdecreasingconcaveup.png|100px]]
|-
| 4 ||[[uses::Local minimum from the right implies right-hand derivative is nonnegative if it exists]] || <math>f</math> has a (possibly, but not necessarily, strict) local minimum from the right at <math>c</math>. || The right-hand derivative <math>f'_+(c)</math>, if it exists, is positive or zero || In the difference quotient, the numerator is positive or zero and the denominator is positive. The quotient is thus positive or zero. Hence, so is the limit, if it exists. || [[File:Rightincreasingconcavedown.png|100px]][[File:Rightincreasingconcaveupnotflat.png|100px]]
|-
|}

The video below provides an intuitive explanation of the above facts. For a full proof, see the page on Fact (1).

<center>{{#widget:YouTube|id=sko6usPekeQ}}</center>

===Note on strictness===

All the facts as stated above are for not necessarily strict one-sided local extrema, i.e., we allow <math>f(x) = f(c)</math> for <math>x</math> on the immediate left or right of <math>c</math>. However, even if we impose the additional condition of strictness, we still cannot eliminate the possibility of the one-sided derivative being zero. The reason is that even though the difference quotient must now be strictly positive or strictly negative in the various cases, the one-sided derivative, which is defined as a one-sided limit of the difference quotient, may still be equal to zero. All the prototypical pictures in the previous section are pictures of strict local extrema, and some of them show a one-sided derivative of zero.

For more, see the note on strictness in the proof for Fact (1).

This observation will be crucial when we piece together the two-sided information.

===Note on sign sensitivity===

In the facts used above, we see that, when ascertaining the sign of the [[difference quotient]] <math>\frac{f(x) - f(c)}{x - c}</math>, the following are true:

* The sign of the numerator is governed by whether the point is a point of local maximum or minimum. For a point of local maximum, the numerator is negative or zero, and for a point of local minimum, the numerator is positive or zero.
* The sign of the denominator is governed by whether we are approaching from the left or the right. For a left-sided approach, the denominator is negative and for a right-sided approach, the denominator is positive.

The upshot of this is that if we change the direction of approach while preserving the nature of the local extreme value, the sign of the one-sided derivative flips. This is crucial to the proof.

==Proof==

===Proof idea===

The idea is to convert ''two-sided'' local extremum information into both its one-sided pieces, then determine the signs of the one-sided derivatives. As noted in [[#Note on sign sensitivity]], the sign conclusions for the two one-sided derivatives are opposite. We then pit these two pieces of information against each other to force the two-sided derivative, if it exists, to equal zero.

===Local maximum case===

'''Given''': A function <math>f</math>, a point <math>c</math> in the interior of the domain of <math>f</math> such that <math>f</math> attains a local maximum at <math>c</math>, i.e., <math>f(x) \le f(c)</math> for all <math>x \in (c - \delta, c + \delta)</math> for some choice of <math>\delta > 0</math>.

'''To prove''': If <math>\! f'(c)</math> exists, then <math>\!f'(c) = 0</math>

'''Proof'''

{| class="sortable" border="1"
! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation
|-
| 1 || If the left-hand derivative of <math>f</math> at <math>c</math> exists, then it is nonnegative, i.e., it is positive or zero. || Fact (1) || <math>f</math> attains a (two-sided) local maximum at <math>c</math> || || Since <math>f</math> attains a ''two-sided'' local maximum at <math>c</math>, it in particular attains a local maximum from the left. Thus, Fact (1) applies.
|-
| 2 || If the right-hand derivative of <math>f</math> at <math>c</math> exists, then it is nonpositive, i.e., it is negative or zero. || Fact (3) || <math>f</math> attains a (two-sided) local maximum at <math>c</math> || || Since <math>f</math> attains a ''two-sided'' local maximum at <math>c</math>, it in particular attains a local maximum from the right. Thus, Fact (3) applies.
|-
| 3 || If the (two-sided) derivative of <math>f</math> exists at <math>c</math>, it must be zero. || || || Steps (1), (2) || For the (two-sided) derivative to exist, ''both'' the left-hand derivative and the right-hand derivative must exist ''and'' they must be equal to each other and to the derivative. By Step (1), the derivative must be positive or zero. By Step (2), the derivative must be negative or zero. The only way both of these can be simultaneously true is if the derivative equals zero.
|}

<center>{{#widget:YouTube|id=gExgdK8xKUs}}</center>

===Local minimum case===

'''Given''': A function <math>f</math>, a point <math>c</math> in the interior of the domain of <math>f</math> such that <math>f</math> attains a local minimum at <math>c</math>, i.e., <math>f(x) \ge f(c)</math> for all <math>x \in (c - \delta, c + \delta)</math> for some choice of <math>\delta > 0</math>.

'''To prove''': If <math>f'(c)</math> exists, then <math>f'(c) = 0</math>

'''Proof'''

{| class="sortable" border="1"
! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation
|-
| 1 || If the left-hand derivative of <math>f</math> at <math>c</math> exists, then it is nonpositive, i.e., it is negative or zero. || Fact (2) || <math>f</math> attains a (two-sided) local minimum at <math>c</math> || || Since <math>f</math> attains a ''two-sided'' local minimum at <math>c</math>, it in particular attains a local minimum from the left. Thus, Fact (2) applies.
|-
| 2 || If the right-hand derivative of <math>f</math> at <math>c</math> exists, then it is nonnegative, i.e., it is positive or zero. || Fact (4) || <math>f</math> attains a (two-sided) local minimum at <math>c</math> || || Since <math>f</math> attains a ''two-sided'' local minimum at <math>c</math>, it in particular attains a local minimum from the right. Thus, Fact (4) applies.
|-
| 3 || If the (two-sided) derivative of <math>f</math> exists at <math>c</math>, it must be zero. || || || Steps (1), (2) || For the (two-sided) derivative to exist, ''both'' the left-hand derivative and the right-hand derivative must exist ''and'' they must be equal to each other and to the derivative. By Step (1), the derivative must be negative or zero. By Step (2), the derivative must be positive or zero. The only way both of these can be simultaneously true is if the derivative equals zero.
|}

===Note on strictness===

Even if we consider the case of ''strict'' two-sided local maximum or strict two-sided local minimum, we can still have either of the two types of critical point: <math>f'(c) = 0</math> or <math>f'(c)</math> not existing. The <math>f'(c) = 0</math> case continues to be possible because, for the one-sided versions, we can have the one-sided derivative equal to zero even assuming ''strict'' one-sided local extremum.

Point of local extremum implies critical point

2024-04-18T06:34:37Z

Vipul: /* Significance */

{{maxmin test}}
==Statement==

Suppose <math>f</math> is a [[function]] of one variable and <math>c</math> is a point in the [[interior]] of the [[domain]] of <math>f</math> (i.e., <math>f</math> is defined on an open interval containing <math>c</math>).

Suppose further that <math>c</math> is a [[point of local extremum]] for <math>f</math>, i.e., <math>f</math> attains a local extreme value (either a local maximum or a local minimum) at <math>c</math>.

Then, <math>c</math> is a [[critical point]] for <math>f</math>, i.e., either the [[derivative]] <math>f'(c)</math> equals zero or the derivative <math>f'(c)</math> does not exist.

Note that the <math>f'(c)</math> not existing case could occur in either of these ways: one or both the one-sided derivatives of <math>f</math> at <math>c</math> not existing, ''or'' both one-sided derivatives of <math>f</math> at <math>c</math> existing but being unequal.

==Significance==

===General comments===

The main significance of this result is that it mostly reduces an ''optimization'' problem (finding a maximum or minimum) to an ''equation-solving'' problem (finding when an expression is zero).

The reduction isn't perfect, in that it has the following caveats:

* The result is only a one-sided implication. Any point of local extremum is a critical point, but it's possible for a critical point to not be a point of local extremum. Even so, in many cases, the set of critical points is already small enough and manageable enough.
* This result is about local extrema, not absolute / global extrema. There may be many points of local extremum that aren't points of absolute extremum. Moreover, there could be many cases where an absolute extremum doesn't exist; for instance, for a function that is going to infinity as the input goes to infinity or to a point of discontunity (such as <math>x \mapsto 1/x</math> as <math>x \to 0^+</math> and <math>x \to 0^-</math>.

===Qualitative and existential significance===

{| class="sortable" border="1"
! Type of function for which we are interested in understanding local extrema !! Inference
|-
| polynomial function of degree <math>d</math> || The derivative is a polynomial of degree <math>d - 1</math>, so it is defined everywhere and has at most <math>d - 1</math> real roots. Therefore, there are at most <math>d - 1</math> critical points (all of the "derivative equals zero" type, none of the "derivative is undefined" type) and therefore at most <math>d - 1</math> points of local extremum for the function.
|-
| rational function in simplified form with numerator degree <math>d_1</math> and denominator degree <math>d_2</math> || The derivative is a rational function with numerator degree at most <math>d_1 + d_2 - 1</math> and denominator a square of the original denominator. The only way for the derivative to be undefined is for the denominator to be zero, in which case the denominator of the original rational function would also be zero, so that the original rational function would also be undefined. Therefore, the derivative exists everywhere that the function is defined, and the only critical points are cases where the derivative is zero. Since the numerator is a polynomial of degree at most <math>d_1 + d_2 - 1</math>, there are therefore at most <math>d_1 + d_2 - 1</math> critical points, and therefore at most <math>d_1 + d_2 - 1</math> points of local extremum.
|}

==Facts used==

===Statement of facts===

{| class="sortable" border="1"
! Fact no. (for reference in proof) !! Statement !! Assumption about one-sided local extremum !! Conclusion about sign of one-sided derivative !! Quick explanation in terms of the [[difference quotient]] <math>\frac{f(x) - f(c)}{x - c}</math> !! Prototypical pictures
|-
| 1 || [[uses::Local maximum from the left implies left-hand derivative is nonnegative if it exists]] (has full proof + video) || <math>f</math> has a (possibly, but not necessarily, strict) local maximum from the left at <math>c</math>. || The left-hand derivative <math>f'_-(c)</math>, if it exists, is positive or zero || In the difference quotient, the numerator is negative or zero and the denominator is negative. The quotient is thus positive or zero. Hence, so is the limit, if it exists. || [[File:Leftincreasingconcaveup.png|100px]][[File:Leftincreasingconcavedownflat.png|100px]][[File:Leftincreasingoscillatoryflat.png|100px]]
|-
| 2 || [[uses::Local minimum from the left implies left-hand derivative is nonpositive if it exists]] || <math>f</math> has a (possibly, but not necessarily, strict) local minimum from the left at <math>c</math>. || The left-hand derivative <math>f'_-(c)</math>, if it exists, is negative or zero || In the difference quotient, the numerator is positive or zero and the denominator is negative. The quotient is thus negative or zero. Hence, so is the limit, if it exists. || [[File:Leftdecreasingconcavedown.png|100px]][[File:Leftdecreasingconcaveupflat.png|100px]]
|-
| 3 || [[uses::Local maximum from the right implies right-hand derivative is nonpositive if it exists]] || <math>f</math> has a (possibly, but not necessarily, strict) local maximum from the right at <math>c</math>. || The right-hand derivative <math>f'_+(c)</math>, if it exists, is negative or zero || In the difference quotient, the numerator is negative or zero and the denominator is positive. The quotient is thus negative or zero. Hence, so is the limit, if it exists. || [[File:Rightdecreasingconcavedownnotflat.png|100px]][[File:Rightdecreasingconcaveup.png|100px]]
|-
| 4 ||[[uses::Local minimum from the right implies right-hand derivative is nonnegative if it exists]] || <math>f</math> has a (possibly, but not necessarily, strict) local minimum from the right at <math>c</math>. || The right-hand derivative <math>f'_+(c)</math>, if it exists, is positive or zero || In the difference quotient, the numerator is positive or zero and the denominator is positive. The quotient is thus positive or zero. Hence, so is the limit, if it exists. || [[File:Rightincreasingconcavedown.png|100px]][[File:Rightincreasingconcaveupnotflat.png|100px]]
|-
|}

The video below provides an intuitive explanation of the above facts. For a full proof, see the page on Fact (1).

<center>{{#widget:YouTube|id=sko6usPekeQ}}</center>

===Note on strictness===

All the facts as stated above are for not necessarily strict one-sided local extrema, i.e., we allow <math>f(x) = f(c)</math> for <math>x</math> on the immediate left or right of <math>c</math>. However, even if we impose the additional condition of strictness, we still cannot eliminate the possibility of the one-sided derivative being zero. The reason is that even though the difference quotient must now be strictly positive or strictly negative in the various cases, the one-sided derivative, which is defined as a one-sided limit of the difference quotient, may still be equal to zero. All the prototypical pictures in the previous section are pictures of strict local extrema, and some of them show a one-sided derivative of zero.

For more, see the note on strictness in the proof for Fact (1).

This observation will be crucial when we piece together the two-sided information.

===Note on sign sensitivity===

In the facts used above, we see that, when ascertaining the sign of the [[difference quotient]] <math>\frac{f(x) - f(c)}{x - c}</math>, the following are true:

* The sign of the numerator is governed by whether the point is a point of local maximum or minimum. For a point of local maximum, the numerator is negative or zero, and for a point of local minimum, the numerator is positive or zero.
* The sign of the denominator is governed by whether we are approaching from the left or the right. For a left-sided approach, the denominator is negative and for a right-sided approach, the denominator is positive.

The upshot of this is that if we change the direction of approach while preserving the nature of the local extreme value, the sign of the one-sided derivative flips. This is crucial to the proof.

==Proof==

===Proof idea===

The idea is to convert ''two-sided'' local extremum information into both its one-sided pieces, then determine the signs of the one-sided derivatives. As noted in [[#Note on sign sensitivity]], the sign conclusions for the two one-sided derivatives are opposite. We then pit these two pieces of information against each other to force the two-sided derivative, if it exists, to equal zero.

===Local maximum case===

'''Given''': A function <math>f</math>, a point <math>c</math> in the interior of the domain of <math>f</math> such that <math>f</math> attains a local maximum at <math>c</math>, i.e., <math>f(x) \le f(c)</math> for all <math>x \in (c - \delta, c + \delta)</math> for some choice of <math>\delta > 0</math>.

'''To prove''': If <math>\! f'(c)</math> exists, then <math>\!f'(c) = 0</math>

'''Proof'''

{| class="sortable" border="1"
! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation
|-
| 1 || If the left-hand derivative of <math>f</math> at <math>c</math> exists, then it is nonnegative, i.e., it is positive or zero. || Fact (1) || <math>f</math> attains a (two-sided) local maximum at <math>c</math> || || Since <math>f</math> attains a ''two-sided'' local maximum at <math>c</math>, it in particular attains a local maximum from the left. Thus, Fact (1) applies.
|-
| 2 || If the right-hand derivative of <math>f</math> at <math>c</math> exists, then it is nonpositive, i.e., it is negative or zero. || Fact (3) || <math>f</math> attains a (two-sided) local maximum at <math>c</math> || || Since <math>f</math> attains a ''two-sided'' local maximum at <math>c</math>, it in particular attains a local maximum from the right. Thus, Fact (3) applies.
|-
| 3 || If the (two-sided) derivative of <math>f</math> exists at <math>c</math>, it must be zero. || || || Steps (1), (2) || For the (two-sided) derivative to exist, ''both'' the left-hand derivative and the right-hand derivative must exist ''and'' they must be equal to each other and to the derivative. By Step (1), the derivative must be positive or zero. By Step (2), the derivative must be negative or zero. The only way both of these can be simultaneously true is if the derivative equals zero.
|}

<center>{{#widget:YouTube|id=gExgdK8xKUs}}</center>

===Local minimum case===

'''Given''': A function <math>f</math>, a point <math>c</math> in the interior of the domain of <math>f</math> such that <math>f</math> attains a local minimum at <math>c</math>, i.e., <math>f(x) \ge f(c)</math> for all <math>x \in (c - \delta, c + \delta)</math> for some choice of <math>\delta > 0</math>.

'''To prove''': If <math>f'(c)</math> exists, then <math>f'(c) = 0</math>

'''Proof'''

{| class="sortable" border="1"
! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation
|-
| 1 || If the left-hand derivative of <math>f</math> at <math>c</math> exists, then it is nonpositive, i.e., it is negative or zero. || Fact (2) || <math>f</math> attains a (two-sided) local minimum at <math>c</math> || || Since <math>f</math> attains a ''two-sided'' local minimum at <math>c</math>, it in particular attains a local minimum from the left. Thus, Fact (2) applies.
|-
| 2 || If the right-hand derivative of <math>f</math> at <math>c</math> exists, then it is nonnegative, i.e., it is positive or zero. || Fact (4) || <math>f</math> attains a (two-sided) local minimum at <math>c</math> || || Since <math>f</math> attains a ''two-sided'' local minimum at <math>c</math>, it in particular attains a local minimum from the right. Thus, Fact (4) applies.
|-
| 3 || If the (two-sided) derivative of <math>f</math> exists at <math>c</math>, it must be zero. || || || Steps (1), (2) || For the (two-sided) derivative to exist, ''both'' the left-hand derivative and the right-hand derivative must exist ''and'' they must be equal to each other and to the derivative. By Step (1), the derivative must be negative or zero. By Step (2), the derivative must be positive or zero. The only way both of these can be simultaneously true is if the derivative equals zero.
|}

===Note on strictness===

Even if we consider the case of ''strict'' two-sided local maximum or strict two-sided local minimum, we can still have either of the two types of critical point: <math>f'(c) = 0</math> or <math>f'(c)</math> not existing. The <math>f'(c) = 0</math> case continues to be possible because, for the one-sided versions, we can have the one-sided derivative equal to zero even assuming ''strict'' one-sided local extremum.

Point of local extremum implies critical point

2024-04-18T06:29:12Z

Vipul:

{{maxmin test}}
==Statement==

Suppose <math>f</math> is a [[function]] of one variable and <math>c</math> is a point in the [[interior]] of the [[domain]] of <math>f</math> (i.e., <math>f</math> is defined on an open interval containing <math>c</math>).

Suppose further that <math>c</math> is a [[point of local extremum]] for <math>f</math>, i.e., <math>f</math> attains a local extreme value (either a local maximum or a local minimum) at <math>c</math>.

Then, <math>c</math> is a [[critical point]] for <math>f</math>, i.e., either the [[derivative]] <math>f'(c)</math> equals zero or the derivative <math>f'(c)</math> does not exist.

Note that the <math>f'(c)</math> not existing case could occur in either of these ways: one or both the one-sided derivatives of <math>f</math> at <math>c</math> not existing, ''or'' both one-sided derivatives of <math>f</math> at <math>c</math> existing but being unequal.

==Significance==

===General comments===

The main significance of this result is that it mostly reduces an ''optimization'' problem (finding a maximum or minimum) to an ''equation-solving'' problem (finding when an expression is zero).

The reduction isn't perfect, in that it has the following caveats:

* The result is only a one-sided implication. Any point of local extremum is a critical point, but it's possible for a critical point to not be a point of local extremum. Even so, in many cases, the set of critical points is already small enough and manageable enough.
* This result is about local extrema, not absolute / global extrema. There may be many points of local extremum that aren't points of absolute extremum. Moreover, there could be many cases where an absolute extremum doesn't exist; for instance, for a function that is going to infinity as the input goes to infinity or to a point of discontunity (such as <math>x \mapsto 1/x</math> as <math>x \to 0^+</math> and <math>x \to 0^-</math>.

===Qualitative and existential significance===

{| class="sortable" border="1"
! Type of function for which we are interested in understanding local extrema !! Inference
|-
| polynomial function of degree <math>d</math> || The derivative is a polynomial of degree <math>d - 1</math>, so it is defined everywhere and has at most <math>d - 1</math> real roots. Therefore, there are at most <math>d - 1</math> critical points (all of the "derivative equals zero" type, none of the "derivative is undefined" type) and therefore at most <math>d - 1</math> points of local extremum for the function.
|-
| rational function with numerator degree <math>d_1</math> and denominator degree <math>d_2</math> || The derivative is a rational function with numerator degree at most <math>d_1 + d_2</math> and denominator a square of a polynomial of degree <math>d_2</math>, so there are at most <math>d_1 + d_2</math> critical points of the "derivative is zero" type and at most <math>d_2</math> critical points of the "derivative is undefined" type. This puts an upper bound of <math>d_1 + 2d_2</math> on the number of critical points.
|}

==Facts used==

===Statement of facts===

{| class="sortable" border="1"
! Fact no. (for reference in proof) !! Statement !! Assumption about one-sided local extremum !! Conclusion about sign of one-sided derivative !! Quick explanation in terms of the [[difference quotient]] <math>\frac{f(x) - f(c)}{x - c}</math> !! Prototypical pictures
|-
| 1 || [[uses::Local maximum from the left implies left-hand derivative is nonnegative if it exists]] (has full proof + video) || <math>f</math> has a (possibly, but not necessarily, strict) local maximum from the left at <math>c</math>. || The left-hand derivative <math>f'_-(c)</math>, if it exists, is positive or zero || In the difference quotient, the numerator is negative or zero and the denominator is negative. The quotient is thus positive or zero. Hence, so is the limit, if it exists. || [[File:Leftincreasingconcaveup.png|100px]][[File:Leftincreasingconcavedownflat.png|100px]][[File:Leftincreasingoscillatoryflat.png|100px]]
|-
| 2 || [[uses::Local minimum from the left implies left-hand derivative is nonpositive if it exists]] || <math>f</math> has a (possibly, but not necessarily, strict) local minimum from the left at <math>c</math>. || The left-hand derivative <math>f'_-(c)</math>, if it exists, is negative or zero || In the difference quotient, the numerator is positive or zero and the denominator is negative. The quotient is thus negative or zero. Hence, so is the limit, if it exists. || [[File:Leftdecreasingconcavedown.png|100px]][[File:Leftdecreasingconcaveupflat.png|100px]]
|-
| 3 || [[uses::Local maximum from the right implies right-hand derivative is nonpositive if it exists]] || <math>f</math> has a (possibly, but not necessarily, strict) local maximum from the right at <math>c</math>. || The right-hand derivative <math>f'_+(c)</math>, if it exists, is negative or zero || In the difference quotient, the numerator is negative or zero and the denominator is positive. The quotient is thus negative or zero. Hence, so is the limit, if it exists. || [[File:Rightdecreasingconcavedownnotflat.png|100px]][[File:Rightdecreasingconcaveup.png|100px]]
|-
| 4 ||[[uses::Local minimum from the right implies right-hand derivative is nonnegative if it exists]] || <math>f</math> has a (possibly, but not necessarily, strict) local minimum from the right at <math>c</math>. || The right-hand derivative <math>f'_+(c)</math>, if it exists, is positive or zero || In the difference quotient, the numerator is positive or zero and the denominator is positive. The quotient is thus positive or zero. Hence, so is the limit, if it exists. || [[File:Rightincreasingconcavedown.png|100px]][[File:Rightincreasingconcaveupnotflat.png|100px]]
|-
|}

The video below provides an intuitive explanation of the above facts. For a full proof, see the page on Fact (1).

<center>{{#widget:YouTube|id=sko6usPekeQ}}</center>

===Note on strictness===

All the facts as stated above are for not necessarily strict one-sided local extrema, i.e., we allow <math>f(x) = f(c)</math> for <math>x</math> on the immediate left or right of <math>c</math>. However, even if we impose the additional condition of strictness, we still cannot eliminate the possibility of the one-sided derivative being zero. The reason is that even though the difference quotient must now be strictly positive or strictly negative in the various cases, the one-sided derivative, which is defined as a one-sided limit of the difference quotient, may still be equal to zero. All the prototypical pictures in the previous section are pictures of strict local extrema, and some of them show a one-sided derivative of zero.

For more, see the note on strictness in the proof for Fact (1).

This observation will be crucial when we piece together the two-sided information.

===Note on sign sensitivity===

In the facts used above, we see that, when ascertaining the sign of the [[difference quotient]] <math>\frac{f(x) - f(c)}{x - c}</math>, the following are true:

* The sign of the numerator is governed by whether the point is a point of local maximum or minimum. For a point of local maximum, the numerator is negative or zero, and for a point of local minimum, the numerator is positive or zero.
* The sign of the denominator is governed by whether we are approaching from the left or the right. For a left-sided approach, the denominator is negative and for a right-sided approach, the denominator is positive.

The upshot of this is that if we change the direction of approach while preserving the nature of the local extreme value, the sign of the one-sided derivative flips. This is crucial to the proof.

==Proof==

===Proof idea===

The idea is to convert ''two-sided'' local extremum information into both its one-sided pieces, then determine the signs of the one-sided derivatives. As noted in [[#Note on sign sensitivity]], the sign conclusions for the two one-sided derivatives are opposite. We then pit these two pieces of information against each other to force the two-sided derivative, if it exists, to equal zero.

===Local maximum case===

'''Given''': A function <math>f</math>, a point <math>c</math> in the interior of the domain of <math>f</math> such that <math>f</math> attains a local maximum at <math>c</math>, i.e., <math>f(x) \le f(c)</math> for all <math>x \in (c - \delta, c + \delta)</math> for some choice of <math>\delta > 0</math>.

'''To prove''': If <math>\! f'(c)</math> exists, then <math>\!f'(c) = 0</math>

'''Proof'''

{| class="sortable" border="1"
! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation
|-
| 1 || If the left-hand derivative of <math>f</math> at <math>c</math> exists, then it is nonnegative, i.e., it is positive or zero. || Fact (1) || <math>f</math> attains a (two-sided) local maximum at <math>c</math> || || Since <math>f</math> attains a ''two-sided'' local maximum at <math>c</math>, it in particular attains a local maximum from the left. Thus, Fact (1) applies.
|-
| 2 || If the right-hand derivative of <math>f</math> at <math>c</math> exists, then it is nonpositive, i.e., it is negative or zero. || Fact (3) || <math>f</math> attains a (two-sided) local maximum at <math>c</math> || || Since <math>f</math> attains a ''two-sided'' local maximum at <math>c</math>, it in particular attains a local maximum from the right. Thus, Fact (3) applies.
|-
| 3 || If the (two-sided) derivative of <math>f</math> exists at <math>c</math>, it must be zero. || || || Steps (1), (2) || For the (two-sided) derivative to exist, ''both'' the left-hand derivative and the right-hand derivative must exist ''and'' they must be equal to each other and to the derivative. By Step (1), the derivative must be positive or zero. By Step (2), the derivative must be negative or zero. The only way both of these can be simultaneously true is if the derivative equals zero.
|}

<center>{{#widget:YouTube|id=gExgdK8xKUs}}</center>

===Local minimum case===

'''Given''': A function <math>f</math>, a point <math>c</math> in the interior of the domain of <math>f</math> such that <math>f</math> attains a local minimum at <math>c</math>, i.e., <math>f(x) \ge f(c)</math> for all <math>x \in (c - \delta, c + \delta)</math> for some choice of <math>\delta > 0</math>.

'''To prove''': If <math>f'(c)</math> exists, then <math>f'(c) = 0</math>

'''Proof'''

{| class="sortable" border="1"
! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation
|-
| 1 || If the left-hand derivative of <math>f</math> at <math>c</math> exists, then it is nonpositive, i.e., it is negative or zero. || Fact (2) || <math>f</math> attains a (two-sided) local minimum at <math>c</math> || || Since <math>f</math> attains a ''two-sided'' local minimum at <math>c</math>, it in particular attains a local minimum from the left. Thus, Fact (2) applies.
|-
| 2 || If the right-hand derivative of <math>f</math> at <math>c</math> exists, then it is nonnegative, i.e., it is positive or zero. || Fact (4) || <math>f</math> attains a (two-sided) local minimum at <math>c</math> || || Since <math>f</math> attains a ''two-sided'' local minimum at <math>c</math>, it in particular attains a local minimum from the right. Thus, Fact (4) applies.
|-
| 3 || If the (two-sided) derivative of <math>f</math> exists at <math>c</math>, it must be zero. || || || Steps (1), (2) || For the (two-sided) derivative to exist, ''both'' the left-hand derivative and the right-hand derivative must exist ''and'' they must be equal to each other and to the derivative. By Step (1), the derivative must be negative or zero. By Step (2), the derivative must be positive or zero. The only way both of these can be simultaneously true is if the derivative equals zero.
|}

===Note on strictness===

Even if we consider the case of ''strict'' two-sided local maximum or strict two-sided local minimum, we can still have either of the two types of critical point: <math>f'(c) = 0</math> or <math>f'(c)</math> not existing. The <math>f'(c) = 0</math> case continues to be possible because, for the one-sided versions, we can have the one-sided derivative equal to zero even assuming ''strict'' one-sided local extremum.

Point of local extremum implies critical point

2024-04-18T06:12:42Z

Vipul: /* Statement */

Point of local extremum implies critical point

2024-04-18T06:12:34Z

Vipul: /* Statement */

Differential equation

2024-04-18T06:11:55Z

Vipul: /* Key observations */

==Definition==

===Formal description===

The term '''differential equation''', sometimes called '''ordinary differential equation''' to distinguish it from [[partial differential equation]]s and other variants, is an equation involving two variables, an ''independent variable'' <math>x</math> and a ''dependent'' variable <math>y</math>, as well as the [[derivative]]s (first and possibly higher) of <math>y</math> with respect to <math>x</math>. Formally, it is an equation of the form:

<math>F(x,y,y',y'',\dots,y^{(k)}) = 0</math>

where <math>F</math> is a function of <math>k + 2</math> variables. Here <math>k \ge 1</math>. Note that <math>F</math> may choose not to use some of the derivatives.

In functional notation, the same differential equation may be written as:

<math>F(x,f(x),f'(x),f''(x),\dots,f^{(k)}(x)) = 0</math>

where <math>f</math> is the function such that <math>y = f(x)</math>.

===Solution concept===

* '''Functional solution''': A function <math>f</math> on the domain of interest is said to be a ''solution'' (or ''functional solution'') to the equation if, when we plug in <math>y = f(x)</math>, the equation holds true for ''all'' <math>x</math> in the domain, i.e.:

<math>F(x,f(x),f'(x),f''(x),\dots,f^{(k)}(x)) = 0 \ \forall \ x \in \operatorname{dom}(f)</math>

Note that in cases of functions defined on closed intervals, we exclude checking the conditions on the boundary of the domain because two-sided derivatives don't make sense at the boundary.

* '''Relational solution''': A relation <math>R(x,y) = 0</math> is termed a ''relational solution'' to the equation if <math>F(x,y,y',y'',\dots,y^{(k)}) = 0</math> holds true for all <math>x,y</math> if we calculate the derivatives of <math>y</math> with respect to <math>x</math> using [[implicit differentiation]].

===Initial value problem===

An ''initial value problem'' is a differential equation:

<math>F(x,y,y',y'',\dots,y^{(k)}) = 0</math>

accompanied with a tuple <math>(x_0,y_0,y_1,\dots,y_{k-1})</math>.

A '''functional solution''' to the initial value problem is a functional solution <math>y = f(x)</math> for the differential equation such that <math>f(x_0) = y_0</math> and <math>f^{(i)}(x_0) = y_i</math> for <math>i \in \{ 1,2,\dots,k-1\}</math>.

Analogously, we can define a relational solution to the initial value problem.

==Key observations==

===Differential equations are functional equations===

Differential equations are examples of [[functional equation]]s. A functional equation is an equation where the ''variable'' that we are trying to solve for is a function, and the equation holds true for all values of the input to the function. For instance, here is an example of a functional equation (that's not a differential equation):

<math>f(x + y) = f(x) + f(y) \ \forall \ x,y \in \R</math>

A ''solution'' to a functional equation is a function that satisfies the equation for all choices of inputs. For instance, any function of the form <math>f(x) := ax</math> for fixed <math>a \in \R</math> is a solution to the above functional equation.

Differential equations are functional equations -- we are trying to solve a differential equation, not for the variables, but for the ''functional'' relationship between them.

===Differential equations capture behavior at a single point===

Not every functional equation involving derivatives is a differential equation. Differential equations are characterized by the evaluation of the function and its derivatives all happening at a single point. For instance:

* <math>x^2 = f(x) + (f'(x))^3</math> is a differential equation because all the function and derivative evaluations happen at <math>x</math>, but
* <math>x^2 = f(x) + f'(1 - x)</math> is ''not'' a differential equation in our sense of the word because the derivative evaluation happens at <math>1 - x</math> rather than <math>x</math>.

Another way of putting this is that differential equations are inherently local and cannot relate the behavior of the function at far-away points.

Functional equations involving derivatives that do ''not'' fit this definition of differential equation are also studied, but the study of these is more complicated and requires new techniques. [[Delay differential equation]]s is one such class of functional equations.

===Functional equations involving iterated derivatives at a single point can be simplied to differential equations===

Consider an equation like this:

<math>(yy''\sin(ye^x + xy'))' = yx^3 + x\cos y</math>

Here, all derivatives are with respect to <math>x</math>.

This does not meet the standard format of a differential equation, because it involves taking a derivative of an intermediate expression <math>yy''\sin(ye^x + xy')</math>, whereas a standard format differential equation only allows for derivatives (including repeated derivatives) of the dependent variable <math>y</math>.

However, this can be ''simplified'' to a differential equation in the standard format, specifically by using various differentiation rules such as the product rule, chain rule, and the linearity of differentiation, plus the formulas for differentiating the sine and exponential functions. Therefore, we generally consider this sort of equation to be an ordinary differential equation, albeit in an unsimplified form.

More abstractly, if <math>G</math> is a function of <math>x, y, y', \dots, y^{(k)}</math>, then <math>G'</math> can be written as follows using [[partial derivative]]s by using the [[chain rule for partial differentiation]]:

<math>G' = G_x + G_y y' + G_{y'} y'' + \dots G_{y^{(k)}} y^{(k+1)}</math>

In particular, this allows us to simplify the intermediate derivative expressions into expressions in <math>x, y, y', \dots, y^{(k + 1)}</math> and thereby allows us to convert the differential equation to a standard format.

(Note that in any particular case such as the example above, we don't need to think in terms of partial derivatives, but the abstract formulation does require the use of partial derivatives).

===It does not make sense to ask whether a point satisfies a differential equation===

Consider a differential equation:

<math>x^2 = y + y'</math>

If I ask the question: ''does the point <math>x = 2,y = 3</math> satisfy the differential equation?'', the answer is that the question doesn't make any sense. This is because verifying a differential equation requires knowing the ''functional relationship'' between <math>x</math> and <math>y</matH>, which in turn allows us to compute the numerical value of <math>y'</math> and check whether the equation is satisfied.

==Terminology==

===Equation terminology===

{| class="sortable" border="1"
! Term !! Meaning !! Example (don't try to solve these differential equations!)
|-
| [[order of a differential equation]] || it is the largest <math>k</math> for which the <math>k^{th}</math> derivative of the dependent variable appears in the differential equation. || The equation <math>y + xy''' + (y'')^2 = \sin(y')</math> has order three because <math>y'''</math> is the largest derivative appearing.
|-
| [[first-order differential equation]] || differential equation of order one, i.e., it involves only <math>x,y,y'</math>. || <math>y' = \sin(x + yy')</math> is a first-order differential equation.
|-
| [[second-order differential equation]] || differential equation of order exactly two, i.e., it involves only <math>x,y,y',y''</math> and has at least one appearance of <math>y''</math>. || <math>xy'' + \cos(y^2y') = x^2e^{yy'}</math> is a second-order differential equation.
|-
| [[degree of a differential equation]] || if the differential equation is polynomial in terms of its highest order derivative, then the degree of that polynomial. || <math>(y''')^2y' + (y')^5 = 3xy</math> has degree two.
|-
| [[explicit differential equation]] || This means that the highest order derivative is written explicitly in terms of the dependent variable, independent variable, and the lower order derivatives. Any explicit differential equation is a first-degree differential equation. Conversely, any first-degree differential equation can be converted to an explicit differential equation by dividing out by the coefficient of the highest order derivative -- if this coefficient is not invertible, we may separately need to consider the case where that coefficient becomes zero, and that would be a ''lower'' order differential equation. || <math>y''' = xy'' - x^2\sin(yy') + y^3(y'')^5</math> is explicit: the third derivative is written in terms of the lower order derivatives.
|-
| [[autonomous differential equation]] || differential equation where the independent variable does ''not'' appear explicitly anywhere in the equation. || <math>y + y'' = \cos(yy'y''')</math> is autonomous. On the other hand, <math>y + xy' = y''</math> is ''not'' autonomous
|-
| [[linear differential equation]] || A differential equation of the form <math>p_k(x)y^{(k)} + p_{k-1}(x)y^{(k-1)} + \dots + p_0(x)y = q(x)</math> where the <math>p_i</math>s and <math>q</math> are all functions. In other words, the expression is linear in <math>y</math> and its derivatives with coefficients in terms of <math>x</math>. Linear differential equations are usually written with the coefficient of <math>y^{(k)}</math> cleared to 1, by dividing throughout by the coefficient of <math>y^{(k)}</math>. || <math>e^xy''' + x^2y'' + \sin(x)y' + 3y = x^2 - 2x + 5</math> is linear.
|-
| [[homogeneous linear differential equation]] || A linear differential equation of the form <math>p_k(x)y^{(k)} + p_{k-1}(x)y^{(k-1)} + \dots + p_0(x)y = 0</math>. In other words, the ''constant term'' function is zero. || <math>e^xy^{(4)} - 3x^3y''' + \sin(\sin x)y = 0</math> is homogeneous linear.
|-
| [[linear differential equation with constant coefficients]] || A linear differential equation of the form <math>a_ky^{(k)} + a_{k-1}y^{(k-1)} + \dots + a_1y' + a_0y = b</math> where all the <math>a_i</math>s and <math>b</math> are zero. || <math>2y^{(5)} - y^{(3)} + y = 13</math> is linear with constant coefficients.
|}

===Solution terminology===

{| class="sortable" border="1"
! Term !! Meaning !! Example
|-
| particular solution || a function or relation that is a solution for the equation (see [[#Solution concept]]). A solution in the form of a function <math>y = f(x)</math> is termed a ''functional solution'' and a solution in the form of a relation <math>R(x,y) = 0</math> is termed a ''relational solution''. || <math>y = \sin x</math> is a functional solution to <math>y^2 + y'^2 = 1</math>.
|-
| solution family || a family of functions or relations, with one or more parameters possibly subject to some constraints, such that for every choice of parameter values subject to those constraints, we get a particular solution. || <math>y = \sin(x + C)</math> with parameter <math>C \in \R</math>, is a solution family for <math>y^2 + y'^2 = 1</math>.
|-
| general solution || a solution family that covers ''all'' solutions (or almost all solutions, possibly excluding some exceptions) || The general solution to <math>y' = 0</math> is <math>y = C, C \in \R</math>.
|-
| solution to initial value problem || a particular solution that satisfies the initial value condition. || A particular solution to <math>y + y' + y'' = (x + 1)^2</math> satisfying <math>y(0) = -1, y'(0) = 0</math> (i.e., <math>y = -1</math> and <math>y' = 0</math> when <math>x = 0</math>) is <math>y = x^2 - 1</math>.
|}

==Facts==

* As a general principle, the way to solve a differential equation of order <math>k</math> is to reduce it to a sequence of <math>k</math> integration problems. Each integration problem introduces a new freely varying parameter.
* As a general principle, the number of degrees of freedom (i.e., the number of independent freely varying parameters) in the general solution to a differential equation of order <math>k</math> must equal <math>k</math>. There are various exceptions and irregularities, but this is what we should generally expect. Another way of putting this is that the solution space to a differential equation of order <math>k</math> is expected to be <math>k</math>-dimensional.
* As a general principle, the number of solutions to an initial value problem should be finite. If the differential equation is nice enough, then there should be a ''unique'' solution to any initial value problem.

==Relation with system of first-order differential equations==

Any differential equation of order <math>k</math> can be converted to a [[system of first-order differential equations]] with <math>k</math> equations and <math>k</math> variables (i.e., <math>k</math> unknown functions that we are trying to solve for). For the conversion procedure, see [[conversion of a differential equation to a system of first-order differential equations]]. However, the converse is not true, i.e., it is not always possible to convert a system of first-order differential equations with multiple dependent variables into a single differential equation of higher order with one dependent variable.

The same idea can be used to perform the [[conversion of a system of differential equations to a system of first-order differential equations]].

==Solution strategies==

===General idea for strategy toward a general solution===

The rough idea is to convert the differential equation to a sequence of integration problems. If the differential equation has order <math>k</math>, it should reduce to performing <math>k</math> integrations. Each integration introduces a new freely varying parameter.

{{quotation|'''CAUTION''': There may be other auxiliary integrations that need to be done, e.g., for computation of integrating factors or in order to solve an [[integration by parts]] problem. These auxiliary antiderivative computations do not, however, introduce new freely varying parameters.}}

We see that this general strategy may run into trouble at many levels:

{| class="sortable" border="1"
! Stage !! Type of difficulty
|-
| Differential equation to integration || There is no general-purpose algorithm for converting an arbitrary differential equation to an integration problem or sequence of integration problems. Thus, we may not even be able to get started on the process. For some types of structures of differential equations, strategies are known for converting them to integration problems. For others, we have to use ''ad hoc'' techniques. There exist differential equations for which there is no way of converting them to integration problems. Note that for differential equations of order two or higher, this problem may occur at ''any'' of the stages, i.e., we may be able to do one level of integration but not the next.
|-
| Solving the integration problem to get rid of the integral sign || Even after we've converted the differential equation to an integration problem, there may not be any analytic methods for computing the antiderivative. Note that this is not such a big issue because there are known techniques for calculating approximate solutions to integration problems even if analytical methods are not available for getting a precise solution.
|-
| Making sense of relational solutions, converting to functional solutions or at least trying to understand what they mean || Even after we have done the desired antidifferentiations, the solutions may be in relational form rather than functional form. This means that we may still not have ''explicit'' functional descriptions of <math>y</math> in terms of <math>x</math>. We may not even know whether <math>y</math> is expressible as a function of <math>x</math>. We may not even know if the relational solution has any points in it at all!
|}

Historically, this type of method of solution of a differential equation is called a ''solution by quadratures''. In the early days of differential equations, it was hoped that generic differential equations could be solved by quadratures, but this hope was dashed fairly quickly.

===Going from a particular solution to a general solution===

There are some special types of differential equations where, once we find a particular solution, we can find other solutions, perhaps even the general solution. Some notable cases are considered here:

* For a [[linear differential equation]], finding a general solution is equivalent to finding a particular solution + solving the corresponding [[homogeneous linear differential equation]]. In particular, for a [[linear differential equation with constant coefficients]], there is a closed form expression for the solution of the corresponding homogeneous linear differential equation with constant coefficients, so finding a particular solution is equivalent to finding the general solution.
* For an [[autonomous differential equation]], if <math>y = f(x)</math> is a solution, so is <math>y = f(x + C)</math> for any constant <math>C</math>. Note that this does not give the ''general solution'' if the order is more than one, but it does help move from a particular solution to a solution family with one parameter.

===Solution strategies in particular cases===

Below are some formats of equations for which general strategies are known. Note that the letter <math>f</math> is no longer used for the solution function but may be used for other functions.:

{| class="sortable" border="1"
! Equation type !! Order !! Degree (if polynomial in highest order derivative) !! Quick summary of solution strategy
|-
| [[first-order linear differential equation]] which in simplified form looks like <math>y' + p(x)y = q(x)</math> || 1 || 1 || Use the [[integrating factor]] <math>e^{H(x)}</math> where <math>H'=p</math>. The general solution is <math>y = Ce^{-H(x)} + e^{-H(x)}\int p(x)e^{H(x)} \, dx</math>
|-
| [[separable differential equation]] which is of the form <math>y' = f(x)g(y)</math> (any [[first-order first-degree autonomous differential equation]] is separable, though there are separable differential equations that aren't autonomous) || 1 || 1 || Separate and solve as <math>\int \frac{dy}{g(y)} = \int f(x) \, dx</math>. Also find solutions corresponding to <math>y = k</math> where <math>g(k) = 0</math>.
|-
| [[first-order exact differential equation]] <math>F(x,y,y') = 0</math> || 1 || 1 || Try to find a relation <math>R(x,y)</math> such that <math>F(x,y,y') = \frac{d}{dx}[R(x,y)]</math> using [[implicit differentiation]]. Finding the <math>R</math>, even if it does exist, can be tricky.
|-
| [[Bernoulli differential equation]] <math>y' + p(x)y = q(x)y^n</math> (<math>n \ne 0,1</math>) || 1 || 1 || Divide both sides by <math>y^n</math> (set aside possible stationary solution <math>y = 0</math>), then substitute <math>w = 1/y^{n-1}</math> to get a [[first-order linear differential equation]] with dependent variable <math>w</math> and independent variable <math>x</math>.
|-
| [[Clairaut's equation]] which is of the form <math>y = xy' + f(y')</math> || 1 || need not be polynomial; if polynomial, may have any degree || <math>y = Cx + f(C)</math> with <math>C \in \R</math> (all straight lines) and a single other solution explicitly described as the solution to <math>x + f(dy/dx) = 0</math>, given by <math>x = -f'(p), y = f(p) - pf'(p)</math> as a parametric curve in terms of <math>p</math>.
|-
| [[Lagrange equation]] <math>y = f(y')x + g(y')</math> which is linear in <math>x</math> and <math>y</math> but not necessarily in <math>y'</math> || 1 || need not be polynomial; if polynomial, may have any degree || General solution is a family of curves, each described as a parametric curve with parameter the derivative <math>y'</math> (which we denote by <math>p</math>). There may be some special straight line solutions of the form <math>y = px + g(p)</math> for values <math>p</math> with <math>p = f(p)</math>.
|-
| [[second-order autonomous differential equation of degree one]], which is of the form <math>y'' = F(y,y')</math> || 2 || 1 ||
|-
| [[homogeneous linear differential equation with constant coefficients]] || any || 1 || We construct the characteristic polynomial of the differential equation, find its real and complex roots, and the space of solution functions is a vector space with basis functions described using these roots.
|}

===Qualitative methods===

For most differential equations, it is very hard to convert the differential equation to a series of integration problems and to find explicit expressions for the solution. Instead, in many cases, we try to determine the qualitative properties of solution functions, including existence, uniqueness, extent of differentiability, nature of roots and critical points, etc.

Product rule for higher derivatives

2024-04-11T06:09:01Z

Vipul: /* Statement */

{{perspectives}}
{{differentiation rule}}

==Statement==

{| class="sortable" border="1"
! Version type !! Statement
|-
| specific point, named functions || This states that if <math>f</math> and <math>g</math> are <math>n</math> times differentiable functions at <math>x = x_0</math>, then the [[pointwise product of functions|pointwise product]] <math>f \cdot g</math> is also <math>n</math> times differentiable at <math>x = x_0</math>, and we have: <math>\frac{d^n}{dx^n}[f(x)g(x)]|_{x = x_0} = \sum_{k=0}^n \binom{n}{k} f^{(n - k)}(x_0)g^{(k)}(x_0)</math> Here, <math>f^{(n - k)}</math> denotes the <math>(n-k)^{th}</math> derivative of <math>f</math> (with <math>f^{(0)} = f, f^{(1)} = f'</math>, etc.), <math>g^{(k)}</math> denotes the <math>k^{th}</math> derivative of <math>g</math>, and <math>\binom{n}{k}</math> is the [[binomial coefficient]]. These are the same as the coefficients that appear in the expansion of <math>(A + B)^n</math>.
|-
| generic point, named functions, point notation || If <math>f</math> and <math>g</math> are functions of one variable, the following holds wherever the right side makes sense: <math>\! \frac{d^n}{dx^n}[f(x)g(x)] = \sum_{k=0}^n \binom{n}{k} f^{(n-k)}(x)g^{(k)}(x)</math>
|-
| generic point, named functions, point-free notation || If <math>f</math> and <math>g</math> are functions of one variable, the following holds wherever the right side makes sense: <math>(fg)^{(n)} = \sum_{k=0}^n \binom{n}{k} f^{(n-k)}g^{(k)}</math>
|-
| Pure Leibniz notation || Suppose <math>u</math> and <math>v</math> are both variables functionally dependent on <math>x</math>. Then <math>\frac{d^n(uv)}{(dx)^n} = \sum_{k=0}^n \binom{n}{k} \frac{d^{n-k}u}{(dx)^{n-k}}\frac{d^kv}{(dx)^k}</math>
|}

===One-sided version===

There are analogues of each of the statements with one-sided derivatives. {{fillin}}

==Particular cases==

{| class="sortable" border="1"
! Value of <math>n</math> !! Formula for <math>\frac{d^n}{dx^n}[f(x)g(x)]</math>
|-
| 1 || <math>\! f'(x)g(x) + f(x)g'(x)</math> (this is the usual [[product rule for differentiation]]).
|-
| 2 || <math>\! f''(x)g(x) + 2f'(x)g'(x) + f(x)g''(x)</math>.
|-
| 3 || <math>\! f'''(x)g(x) + 3f''(x)g'(x) + 3f'(x)g''(x) + g'''(x)</math>.
|-
| 4 || <math>\! f^{(4)}(x)g(x) + 4f'''(x)g'(x) + 6f''(x)g''(x) + 4f'(x)g'''(x) + f(x)g^{(4)}(x)</math>
|-
| 5 || <math>\! f^{(5)}(x)g(x) + 5f^{(4)}(x) g'(x) + 10f'''(x)g''(x) + 10f''(x)g'''(x) + 5f(x)g^{(4)}(x) + g^{(5)}(x)</math>
|}

==Related rules==

===Similar rules in single variable calculus===

* [[Product rule for differentiation]]
* [[Chain rule for higher derivatives]]
* [[Chain rule for differentiation]]
* [[Repeated differentiation is linear]]

===Similar rules in multivariable calculus===

* [[Product rule for higher partial derivatives]]

===Similar rules in advanced mathematics===

* [[Groupprops:Binomial formula for powers of a derivation|Binomial formula for powers of a derivation]]

==Significance==

===Qualitative and existential significance===

Each of the versions has its own qualitative significance:

{| class="sortable" border="1"
! Version type !! Significance
|-
| specific point, named functions || This tells us that if <math>f</math> and <math>g</math> are both <math>n</math> times differentiable at a point <math>x_0</math>, so is <math>f \cdot g</math>.
|-
| generic point, named functions, point notation || This tells us that if <math>f</math> and <math>g</math> are both <math>n</math> times differentiable on an [[open interval]], so is <math>f \cdot g</math>.
|-
| generic point, named functions, point-free notation || This shows that the way that <math>(f \cdot g)^{(n)}</math> behaves is governed by the nature of the derivatives (up to the <math>n^{th}</math>) of <math>f</math> and <math>g</math>. In particular, if <math>f^{(n)}</math> and <math>g^{(n)}</math> are both continuous functions on an interval, so is <math>(f \cdot g)^{(n)}</math>.
|}

===Computational feasibility significance===

Each of the versions has its own computational feasibility significance:

{| class="sortable" border="1"
! Version type !! Significance
|-
| specific point, named functions || This tells us that knowing the values (in the sense of ''numerical values'') of <math>f,f',f'',\dots, f^{(n)}</math> and <math>g,g',g'',\dots,g^{(n)}</math> at a point <math>x_0</math> allows us to compute the value <math>(f \cdot g)^{(n)}(x_0)</math> by plugging into the formula and doing a bunch of multiplications and additions.
|-
| generic point, named functions || This tells us that knowledge of the ''generic'' expressions for <math>f,f',f'',\dots, f^{(n)}</math> and <math>g,g',g'',\dots,g^{(n)}</math> allows us to compute the generic expression for <math>(f \cdot g)^{(n)}</math>.
|}

Product rule for higher derivatives

2024-04-11T06:08:21Z

Vipul: /* Statement */

{{perspectives}}
{{differentiation rule}}

==Statement==

{| class="sortable" border="1"
! Version type !! Statement
|-
| specific point, named functions || This states that if <math>f</math> and <math>g</math> are <math>n</math> times differentiable functions at <math>x = x_0</math>, then the [[pointwise product of functions|pointwise product]] <math>f \cdot g</math> is also <math>n</math> times differentiable at <math>x = x_0</math>, and we have: <math>\frac{d^n}{dx^n}[f(x)g(x)]|_{x = x_0} = \sum_{k=0}^n \binom{n}{k} f^{(n - k)}(x_0)g^{(k)}(x_0)</math> Here, <math>f^{(n - k)}</math> denotes the <math>(n-k)^{th}</math> derivative of <math>f</math> (with <math>f^{(0)} = f, f^{(1)} = f'</math>, etc.), <math>g^{(k)}</math> denotes the <math>k^{th}</math> derivative of <math>g</math>, and <math>\binom{n}{k}</math> is the [[binomial coefficient]]. These are the same as the coefficients that appear in the expansion of <math>(A + B)^n</math>.
|-
| generic point, named functions, point notation || If <math>f</math> and <math>g</math> are functions of one variable, the following holds wherever the right side makes sense: <math>\! \frac{d^n}{dx^n}[f(x)g(x)] = \sum_{k=0}^n \binom{n}{k} f^{(n-k)}(x)g^{(k)}(x)</math>
|-
| generic point, named functions, point-free notation || If <math>f</math> and <math>g</math> are functions of one variable, the following holds wherever the right side makes sense: <math>\! (fg)^{(n)} = \sum_{k=0}^n \binom{n}{k} f^{(n-k)}g^{(k)}</math>
|-
| Pure Leibniz notation || Suppose <math>u</math> and <math>v</math> are both variables functionally dependent on <math>x</math>. Then <math>\frac{d^n(uv)}{(dx)^n} = \sum_{k=0}^n \binom{n}{k} \frac{d^{n-k}u}{(dx)^{n-k}}\frac{d^kv}{(dx)^k}</math>
|}

===One-sided version===

There are analogues of each of the statements with one-sided derivatives. {{fillin}}

==Particular cases==

{| class="sortable" border="1"
! Value of <math>n</math> !! Formula for <math>\frac{d^n}{dx^n}[f(x)g(x)]</math>
|-
| 1 || <math>\! f'(x)g(x) + f(x)g'(x)</math> (this is the usual [[product rule for differentiation]]).
|-
| 2 || <math>\! f''(x)g(x) + 2f'(x)g'(x) + f(x)g''(x)</math>.
|-
| 3 || <math>\! f'''(x)g(x) + 3f''(x)g'(x) + 3f'(x)g''(x) + g'''(x)</math>.
|-
| 4 || <math>\! f^{(4)}(x)g(x) + 4f'''(x)g'(x) + 6f''(x)g''(x) + 4f'(x)g'''(x) + f(x)g^{(4)}(x)</math>
|-
| 5 || <math>\! f^{(5)}(x)g(x) + 5f^{(4)}(x) g'(x) + 10f'''(x)g''(x) + 10f''(x)g'''(x) + 5f(x)g^{(4)}(x) + g^{(5)}(x)</math>
|}

==Related rules==

===Similar rules in single variable calculus===

* [[Product rule for differentiation]]
* [[Chain rule for higher derivatives]]
* [[Chain rule for differentiation]]
* [[Repeated differentiation is linear]]

===Similar rules in multivariable calculus===

* [[Product rule for higher partial derivatives]]

===Similar rules in advanced mathematics===

* [[Groupprops:Binomial formula for powers of a derivation|Binomial formula for powers of a derivation]]

==Significance==

===Qualitative and existential significance===

Each of the versions has its own qualitative significance:

{| class="sortable" border="1"
! Version type !! Significance
|-
| specific point, named functions || This tells us that if <math>f</math> and <math>g</math> are both <math>n</math> times differentiable at a point <math>x_0</math>, so is <math>f \cdot g</math>.
|-
| generic point, named functions, point notation || This tells us that if <math>f</math> and <math>g</math> are both <math>n</math> times differentiable on an [[open interval]], so is <math>f \cdot g</math>.
|-
| generic point, named functions, point-free notation || This shows that the way that <math>(f \cdot g)^{(n)}</math> behaves is governed by the nature of the derivatives (up to the <math>n^{th}</math>) of <math>f</math> and <math>g</math>. In particular, if <math>f^{(n)}</math> and <math>g^{(n)}</math> are both continuous functions on an interval, so is <math>(f \cdot g)^{(n)}</math>.
|}

===Computational feasibility significance===

Each of the versions has its own computational feasibility significance:

{| class="sortable" border="1"
! Version type !! Significance
|-
| specific point, named functions || This tells us that knowing the values (in the sense of ''numerical values'') of <math>f,f',f'',\dots, f^{(n)}</math> and <math>g,g',g'',\dots,g^{(n)}</math> at a point <math>x_0</math> allows us to compute the value <math>(f \cdot g)^{(n)}(x_0)</math> by plugging into the formula and doing a bunch of multiplications and additions.
|-
| generic point, named functions || This tells us that knowledge of the ''generic'' expressions for <math>f,f',f'',\dots, f^{(n)}</math> and <math>g,g',g'',\dots,g^{(n)}</math> allows us to compute the generic expression for <math>(f \cdot g)^{(n)}</math>.
|}

Quiz:Product rule for higher derivatives

2024-04-11T06:06:23Z

Vipul: /* Practical */

{{perspectives}}

For a quiz that tests all the differentiation rules together, see [[Quiz:Differentiation rules]].

==Practical==

<quiz display=simple>
{Suppose <math>f</math> and <math>g</math> are continuous functions at <math>x_0</math>. Suppose <math>f(x_0) = f'(x_0) = f''(x_0) = f'''(x_0) = g(x_0) = g'(x_0) = g''(x_0) = g'''(x_0) = 1</math>. What is <math>(f \cdot g)'''(x_0)</math>?
|type="()"}
- 1
- 2
- 4
- 6
+ 8
</quiz>

==Significance==

===Qualitative and existential significance===

<quiz display=simple>
{Suppose <math>f</math> and <math>g</math> are continuous functions at <math>x_0</math>. Suppose we know that <math>f</math> is 5 times differentiable at <math>x_0</math> and <math>g</math> is 3 times differentiable at <math>x_0</math>. What is the maximum number of times we can be sure (from this information) that <math>f \cdot g</math> is differentiable at <math>x_0</math>?
|type="()"}
+ 3
- 5
- 8
- 15
</quiz>

Quiz:Product rule for higher derivatives

2024-04-11T06:04:17Z

Vipul: /* Practical */

{{perspectives}}

For a quiz that tests all the differentiation rules together, see [[Quiz:Differentiation rules]].

==Practical==

<quiz display=simple>
{Suppose <math>f</math> and <math>g</math> are continuous functions at <math>x_0</math>. Suppose we know that <math>f</math> is 5 times differentiable at <math>x_0</math> and <math>g</math> is 3 times differentiable at <math>x_0</math>. What is the maximum number of times we can be sure (from this information) that <math>f \cdot g</math> is differentiable at <math>x_0</math>?
|type="()"}
+ 3
- 5
- 8
- 15

{Suppose <math>f</math> and <math>g</math> are continuous functions at <math>x_0</math>. Suppose <math>f(x_0) = f'(x_0) = f''(x_0) = f'''(x_0) = g(x_0) = g'(x_0) = g''(x_0) = g'''(x_0) = 1</math>. What is <math>(f \cdot g)'''(x_0)</math>?
|type="()"}
- 1
- 2
- 4
- 6
+ 8
</quiz>

Product rule for higher derivatives

2024-04-11T05:59:45Z

Vipul:

{{perspectives}}
{{differentiation rule}}

==Statement==

{| class="sortable" border="1"
! Version type !! Statement
|-
| specific point, named functions || This states that if <math>f</math> and <math>g</math> are <math>n</math> times differentiable functions at <math>x = x_0</math>, then the [[pointwise product of functions|pointwise product]] <math>f \cdot g</math> is also <math>n</math> times differentiable at <math>x = x_0</math>, and we have: <math>\frac{d^n}{dx^n}[f(x)g(x)]|_{x = x_0} = \sum_{k=0}^n \binom{n}{k} f^{(n - k)}(x_0)g^{(k)}(x_0)</math> Here, <math>f^{(n - k)}</math> denotes the <math>(n-k)^{th}</math> derivative of <math>f</math> (with <math>f^{(0)} = f, f^{(1)} = f'</math>, etc.), <math>g^{(k)}</math> denotes the <math>k^{th}</math> derivative of <math>g</math>, and <math>\binom{n}{k}</math> is the [[binomial coefficient]]. These are the same as the coefficients that appear in the expansion of <math>\! (A + B)^n</math>.
|-
| generic point, named functions, point notation || If <math>f</math> and <math>g</math> are functions of one variable, the following holds wherever the right side makes sense: <math>\! \frac{d^n}{dx^n}[f(x)g(x)] = \sum_{k=0}^n \binom{n}{k} f^{(n-k)}(x)g^{(k)}(x)</math>
|-
| generic point, named functions, point-free notation || If <math>f</math> and <math>g</math> are functions of one variable, the following holds wherever the right side makes sense: <math>\! (fg)^{(n)} = \sum_{k=0}^n \binom{n}{k} f^{(n-k)}g^{(k)}</math>
|-
| Pure Leibniz notation || Suppose <math>u</math> and <math>v</math> are both variables functionally dependent on <math>x</math>. Then <math>\frac{d^n(uv)}{(dx)^n} = \sum_{k=0}^n \binom{n}{k} \frac{d^{n-k}u}{(dx)^{n-k}}\frac{d^kv}{(dx)^k}</math>
|}

===One-sided version===

There are analogues of each of the statements with one-sided derivatives. {{fillin}}

==Particular cases==

{| class="sortable" border="1"
! Value of <math>n</math> !! Formula for <math>\frac{d^n}{dx^n}[f(x)g(x)]</math>
|-
| 1 || <math>\! f'(x)g(x) + f(x)g'(x)</math> (this is the usual [[product rule for differentiation]]).
|-
| 2 || <math>\! f''(x)g(x) + 2f'(x)g'(x) + f(x)g''(x)</math>.
|-
| 3 || <math>\! f'''(x)g(x) + 3f''(x)g'(x) + 3f'(x)g''(x) + g'''(x)</math>.
|-
| 4 || <math>\! f^{(4)}(x)g(x) + 4f'''(x)g'(x) + 6f''(x)g''(x) + 4f'(x)g'''(x) + f(x)g^{(4)}(x)</math>
|-
| 5 || <math>\! f^{(5)}(x)g(x) + 5f^{(4)}(x) g'(x) + 10f'''(x)g''(x) + 10f''(x)g'''(x) + 5f(x)g^{(4)}(x) + g^{(5)}(x)</math>
|}

==Related rules==

===Similar rules in single variable calculus===

* [[Product rule for differentiation]]
* [[Chain rule for higher derivatives]]
* [[Chain rule for differentiation]]
* [[Repeated differentiation is linear]]

===Similar rules in multivariable calculus===

* [[Product rule for higher partial derivatives]]

===Similar rules in advanced mathematics===

* [[Groupprops:Binomial formula for powers of a derivation|Binomial formula for powers of a derivation]]

==Significance==

===Qualitative and existential significance===

Each of the versions has its own qualitative significance:

{| class="sortable" border="1"
! Version type !! Significance
|-
| specific point, named functions || This tells us that if <math>f</math> and <math>g</math> are both <math>n</math> times differentiable at a point <math>x_0</math>, so is <math>f \cdot g</math>.
|-
| generic point, named functions, point notation || This tells us that if <math>f</math> and <math>g</math> are both <math>n</math> times differentiable on an [[open interval]], so is <math>f \cdot g</math>.
|-
| generic point, named functions, point-free notation || This shows that the way that <math>(f \cdot g)^{(n)}</math> behaves is governed by the nature of the derivatives (up to the <math>n^{th}</math>) of <math>f</math> and <math>g</math>. In particular, if <math>f^{(n)}</math> and <math>g^{(n)}</math> are both continuous functions on an interval, so is <math>(f \cdot g)^{(n)}</math>.
|}

===Computational feasibility significance===

Each of the versions has its own computational feasibility significance:

{| class="sortable" border="1"
! Version type !! Significance
|-
| specific point, named functions || This tells us that knowing the values (in the sense of ''numerical values'') of <math>f,f',f'',\dots, f^{(n)}</math> and <math>g,g',g'',\dots,g^{(n)}</math> at a point <math>x_0</math> allows us to compute the value <math>(f \cdot g)^{(n)}(x_0)</math> by plugging into the formula and doing a bunch of multiplications and additions.
|-
| generic point, named functions || This tells us that knowledge of the ''generic'' expressions for <math>f,f',f'',\dots, f^{(n)}</math> and <math>g,g',g'',\dots,g^{(n)}</math> allows us to compute the generic expression for <math>(f \cdot g)^{(n)}</math>.
|}

Quiz:Product rule for higher derivatives

2024-04-11T05:51:14Z

Vipul: Created page with "{{perspectives}} For a quiz that tests all the differentiation rules together, see Quiz:Differentiation rules. ==Practical== <quiz display=simple> {Suppose <math>f</ma..."

Quiz:Product rule for differentiation

2024-04-11T05:44:29Z

Vipul: /* Qualitative and existential significance */

{{perspectives}}

For a quiz that tests all the differentiation rules together, see [[Quiz:Differentiation rules]].

==Practical==

Corresponds to [[Practical:Product rule for differentiation]].

General difficulty level of questions in this section: School level (unless otherwise specified).
<quiz display=simple>
{Suppose <math>f</math> and <math>g</math> are both defined and differentiable at the point 1. Suppose <math>\! f(1) = 2, g(1) = 5, f'(1) = 4, g'(1) = 11</math>. What is the value of <math>(f \cdot g)'(1)</math> where <math>f \cdot g</math> denotes the [[pointwise product of functions]]?
|type="()"}
+ 42
- 44
- 54
- 63
- The information given is insufficient to find <math>(f \cdot g)'(1)</math>.

{What is the derivative of the function <math>x \mapsto \exp(x) \sin x</math>? Hint for derivatives of individual functions: <toggledisplay>Derivative of exponential function is exponential function, derivative of sine function is cosine function.</toggledisplay>
|type="()"}
+ <math>x \mapsto \exp(x)(\sin x + \cos x)</math>
- <math>x \mapsto \exp(x)(\cos x - \sin x)</math>
- <math>x \mapsto \exp(x)(\sin x - \cos x)</math>
- <math>x \mapsto \exp(x)\cos x + \exp(1) \sin x</math>
- <math>x \mapsto \exp(x)\sin x + \exp(1) \cos x</math>

{What is the derivative of the function <math>x \mapsto \sqrt{x}\sin(x^2)</math> for <math>x > 0</math>? This question also requires use of [[chain rule for differentiation]].
|type="()"}
- <math>x \mapsto \cos(x^2)/(2\sqrt{x})</math>
- <math>x \mapsto 2\sqrt{x}\cos(x^2)</math>
- <math>x \mapsto 2\sqrt{x}(\cos(x^2 + \sin(x^2))</math>
- <math>x \mapsto 2x\sqrt{x}\sin(x^2) + \cos(x^2)/(2 \sqrt{x})</math>
+ <math>x \mapsto 2x\sqrt{x}\cos(x^2) + \sin(x^2)/(2 \sqrt{x})</math>

{What is the derivative of the function <math>x \mapsto x \sin x \ln x</math> for <math>x > 0</math>? Hint for derivatives of individual functions: <toggledisplay>Derivative of <math>\sin</math> is <math>\cos</math>, derivative of <math>\ln</math> is <math>x \mapsto 1/x</math></toggledisplay>
|type="()"}
- <math>x \mapsto (\cos x)/x</math>
- <math>x \mapsto (-\cos x)/x</math>
- <math>x \mapsto \cos x \ln x + \cos x + (\sin x)/x</math>
- <math>x \mapsto \cos x \ln x - \cos x + (\sin x)/x</math>
+ <math>x \mapsto \sin x \ln x + x \cos x \ln x + \sin x</math>

</quiz>

==Formulas==

General difficulty level of questions in this section: College level (unless otherwise specified)

<quiz display=simple>
{Suppose <math>f</math> and <math>g</math> are both twice differentiable functions everywhere on <math>\R</math>. Which of the following is the correct formula for <math>(f \cdot g)''</math>, the second derivative of the [[pointwise product of functions]]?
|type="()"}
- <math>f'' \cdot g + f \cdot g''</math>
- <math>f'' \cdot g + f' \cdot g' + f \cdot g''</math>
+ <math>f'' \cdot g + 2f' \cdot g' + f \cdot g''</math>
|| See [[product rule for differentiation]], [[product rule for higher derivatives]].
- <math>f'' \cdot g - f' \cdot g' + f \cdot g''</math>
- <math>f'' \cdot g - 2f' \cdot g' + f \cdot g''</math>

{ Suppose <math>f_1,f_2,f_3</math> are everywhere differentiable functions from <math>\R</math> to <math>\R</math>. What is the derivative <math>(f_1 \cdot f_2 \cdot f_3)'</math>, where <math>f_1 \cdot f_2 \cdot f_3</math> denotes the [[pointwise product of functions]]?
|type="()"}
- <math>f_1' \cdot f_2' \cdot f_3'</math>
+ <math>f_1' \cdot f_2 \cdot f_3 + f_1 \cdot f_2' \cdot f_3 + f_1 \cdot f_2 \cdot f_3'</math>
|| See [[product rule for differentiation#Statement for multiple functions]]
- <math>f_1 \cdot f_2' \cdot f_3' + f_1' \cdot f_2 \cdot f_3' + f_1' \cdot f_2' \cdot f_3</math>
- <math>f_1' \cdot f_2 + f_2' \cdot f_3 + f_3' \cdot f_1</math>
- <math>f_1'' \cdot f_2' \cdot f_3</math>
</quiz>

==Significance==

===Qualitative and existential significance===

Corresponds to [[Product rule for differentiation#Qualitative and existential significance]].

General difficulty level of questions in this section: College level (unless otherwise specified).

<quiz display=simple>
{Suppose <math>f</math> and <math>g</math> are continuous functions at <math>x_0</math> and <math>f \cdot g</math> is the [[pointwise product of functions]]. Which of the following is ''true'' (see last two options!)?
|type="()"}
- If <math>f</math> and <math>g</math> are both left differentiable at <math>x_0</math>, then so is <math>f \cdot g</math>.
- If <math>f</math> and <math>g</math> are both right differentiable at <math>x_0</math>, then so is <math>f \cdot g</math>.
- If <math>f</math> and <math>g</math> are both differentiable at <math>x_0</math>, then so is <math>f \cdot g</math>.
+ All of the above are true
- None of the above is true

{Suppose <math>f</math> and <math>g</math> are continuous functions at <math>x_0</math> and <math>f \cdot g</math> is the [[pointwise product of functions]]. What is the relationship between the differentiability of <math>f</math>, <math>g</math>, and <math>f \cdot g</math> at <math>x_0</math>?
|type="()"}
- If any two of the three functions are differentiable at <math>x_0</math>, then so is the third.
- If <math>f \cdot g</math> is differentiable at <math>x_0</math>, so are <math>f</math> and <math>g</math>.
- If <math>f \cdot g</math> and <math>f</math> are differentiable at <math>x_0</math>, so is <math>g</math>. However, differentiability of <math>f</math> and <math>g</math> at <math>x_0</math> does not guarantee differentiability of <math>f \cdot g</math>.
+ If <math>f</math> and <math>g</math> are both differentiable at <math>x_0</math>, so is <math>f \cdot g</math>. However, differentiability of <math>f \cdot g</math> and <math>f</math> does not guarantee differentiability of <math>g</math>, and differentiability of <math>f \cdot g</math> and <math>g</math> does not guarantee differentiability of <math>f</math>.
- We cannot draw any inferences about differentiability of one of the three functions based on differentiability of the other two.

{Suppose <math>f</math> and <math>g</math> are continuous functions defined on all of <math>\R</math>. Suppose <math>A</math> is the subset of <math>\R</math> comprising those points where <math>f</math> is differentiable, and <matH>B</math> is the subset of <math>\R</math> comprising those points where <math>g</math> is differentiable. Then, what can we say is '''definitely true''' about the subset of <math>\R</math> comprising those points where the [[pointwise product of functions]] <math>f \cdot g</math> is differentiable?
|type="()"}
- It is contained in the intersection <math>A \cap B</math>
+ It contains the intersection <math>A \cap B</math>
- It is contained in the union <math>A \cup B</math>
- It contains the union <math>A \cup B</math>
- None of the above

{Suppose <math>f</math> and <math>g</math> are continuous functions defined on all of <math>\R</math>. Suppose <math>a,b \in \R</math> are such that <math>1 < a < b</math>. Suppose <math>f</math> is known to be differentiable at <math>a</math> and <math>g</math> is known to be differentiable at <math>b</math>. We do not have information about where else <math>f</math> or <math>g</math> is differentiable. What can we conclude about where the [[pointwise product of functions]] <math>f \cdot g</math> is differentiable?
|type="()"}
- <math>f \cdot g</math> is differentiable at <math>ab</math>
- <math>f \cdot g</math> is differentiable at at least one of <math>a</math> or <math>b</math> but not necessarily at both
- <math>f \cdot g</math> is differentiable at both <math>a</math> and <math>b</math>
+ We don't have enough information to conclude anything about the set of points where <math>f \cdot g</math> is differentiable

{Suppose <math>\mathcal{F}</math> is a collection of differentiable functions defined on all of <math>\R</math>. Further, suppose that there is a collection <math>\mathcal{B}</math> of functions such that every element of <math>\mathcal{F}</math> can be written as a polynomial in terms of the elements of <math>\mathcal{B}</math>, with constant coefficients. Suppose that the derivative of every element of <math>\mathcal{B}</math> is in <math>\mathcal{F}</math>. Which of the following conditions are sufficient to ensure that the derivative of every element of <math>\mathcal{F}</math> is in <math>\mathcal{F}</math>?
|type="()"}
- It is sufficient to ensure that <math>\mathcal{F}</math> is closed under addition and scalar multiplication, i.e., it forms a [[vector space]] of functions.
- It is sufficient to ensure that <math>\mathcal{F}</math> is closed under multiplication, i.e., the product of any two elements of <math>\mathcal{F}</math> is in <math>\mathcal{F}</math>.
+ It is sufficient to ensure that <math>\mathcal{F}</math> is closed under addition-cum-scalar multiplication (i.e., it forms a vector space), ''and'' closed under multiplication, but just having one of those conditions need not suffice.
- It is not sufficient to ensure that <math>\mathcal{F}</math> is closed under addition-cum-scalar multiplication (i.e., it forms a vector space), ''and'' closed under multiplication
</quiz>

===Computational feasibility significance===

See the section [[#Practical]].

===Computational results significance===

Corresponds to [[Product rule for differentiation#Computational results significance]].

General difficulty level of questions in this section: College level (unless otherwise specified).

<quiz display=simple>
{Suppose <math>f</math> and <math>g</math> are infinitely differentiable functions on all of <math>\R</math> such that both <math>f'</math> and <math>g'</math> are [[periodic function]]s with the same period <math>h > 0</math>. What can we conclude about <math>f \cdot g</math>?
|type="()"}
- <math>f \cdot g</math> must be periodic
- <math>(f \cdot g)'</math> must be periodic, but <math>f \cdot g</math> may or may not be periodic.
- <math>(f \cdot g)''</math> must be periodic, but <math>(f \cdot g)'</math> may or may not be periodic.
- <math>(f \cdot g)'''</math> must be periodic, but <math>(f \cdot g)''</math> may or may not be periodic.
+ We cannot conclude from the given information whether <math>f \cdot g</math> or any of the derivatives of <math>f \cdot g</math> is periodic.

{Suppose <math>f</math> and <math>g</math> are functions defined and differentiable on the open interval <math>(0,1)</math>. Suppose, further, that on <math>(0,1)</math>, the derivative functions <math>f'</math> and <math>g'</math> are both expressible as [[rational function]]s. What can we say about <math>f \cdot g</math> and <math>(f \cdot g)'</math> on <math>(0,1)</math>?
|type="()"}
- Both <math>f \cdot g</math> and <math>(f \cdot g)'</math> are expressible as rational functions.
- <math>f \cdot g</math> is expressible as a rational function, but <math>(f \cdot g)'</math> need not be expressible as a rational function.
- <math>(f \cdot g)'</math> is expressible as a rational function, but <math>f \cdot g</math> need not be expressible as a rational function.
+ Neither <math>f \cdot g</math> nor <math>(f \cdot g)'</math> need be expressible as a rational function.
</quiz>

Quiz:Product rule for differentiation

2024-04-11T05:43:33Z

Vipul: /* Qualitative and existential significance */

{{perspectives}}

For a quiz that tests all the differentiation rules together, see [[Quiz:Differentiation rules]].

==Practical==

Corresponds to [[Practical:Product rule for differentiation]].

General difficulty level of questions in this section: School level (unless otherwise specified).
<quiz display=simple>
{Suppose <math>f</math> and <math>g</math> are both defined and differentiable at the point 1. Suppose <math>\! f(1) = 2, g(1) = 5, f'(1) = 4, g'(1) = 11</math>. What is the value of <math>(f \cdot g)'(1)</math> where <math>f \cdot g</math> denotes the [[pointwise product of functions]]?
|type="()"}
+ 42
- 44
- 54
- 63
- The information given is insufficient to find <math>(f \cdot g)'(1)</math>.

{What is the derivative of the function <math>x \mapsto \exp(x) \sin x</math>? Hint for derivatives of individual functions: <toggledisplay>Derivative of exponential function is exponential function, derivative of sine function is cosine function.</toggledisplay>
|type="()"}
+ <math>x \mapsto \exp(x)(\sin x + \cos x)</math>
- <math>x \mapsto \exp(x)(\cos x - \sin x)</math>
- <math>x \mapsto \exp(x)(\sin x - \cos x)</math>
- <math>x \mapsto \exp(x)\cos x + \exp(1) \sin x</math>
- <math>x \mapsto \exp(x)\sin x + \exp(1) \cos x</math>

{What is the derivative of the function <math>x \mapsto \sqrt{x}\sin(x^2)</math> for <math>x > 0</math>? This question also requires use of [[chain rule for differentiation]].
|type="()"}
- <math>x \mapsto \cos(x^2)/(2\sqrt{x})</math>
- <math>x \mapsto 2\sqrt{x}\cos(x^2)</math>
- <math>x \mapsto 2\sqrt{x}(\cos(x^2 + \sin(x^2))</math>
- <math>x \mapsto 2x\sqrt{x}\sin(x^2) + \cos(x^2)/(2 \sqrt{x})</math>
+ <math>x \mapsto 2x\sqrt{x}\cos(x^2) + \sin(x^2)/(2 \sqrt{x})</math>

{What is the derivative of the function <math>x \mapsto x \sin x \ln x</math> for <math>x > 0</math>? Hint for derivatives of individual functions: <toggledisplay>Derivative of <math>\sin</math> is <math>\cos</math>, derivative of <math>\ln</math> is <math>x \mapsto 1/x</math></toggledisplay>
|type="()"}
- <math>x \mapsto (\cos x)/x</math>
- <math>x \mapsto (-\cos x)/x</math>
- <math>x \mapsto \cos x \ln x + \cos x + (\sin x)/x</math>
- <math>x \mapsto \cos x \ln x - \cos x + (\sin x)/x</math>
+ <math>x \mapsto \sin x \ln x + x \cos x \ln x + \sin x</math>

</quiz>

==Formulas==

General difficulty level of questions in this section: College level (unless otherwise specified)

<quiz display=simple>
{Suppose <math>f</math> and <math>g</math> are both twice differentiable functions everywhere on <math>\R</math>. Which of the following is the correct formula for <math>(f \cdot g)''</math>, the second derivative of the [[pointwise product of functions]]?
|type="()"}
- <math>f'' \cdot g + f \cdot g''</math>
- <math>f'' \cdot g + f' \cdot g' + f \cdot g''</math>
+ <math>f'' \cdot g + 2f' \cdot g' + f \cdot g''</math>
|| See [[product rule for differentiation]], [[product rule for higher derivatives]].
- <math>f'' \cdot g - f' \cdot g' + f \cdot g''</math>
- <math>f'' \cdot g - 2f' \cdot g' + f \cdot g''</math>

{ Suppose <math>f_1,f_2,f_3</math> are everywhere differentiable functions from <math>\R</math> to <math>\R</math>. What is the derivative <math>(f_1 \cdot f_2 \cdot f_3)'</math>, where <math>f_1 \cdot f_2 \cdot f_3</math> denotes the [[pointwise product of functions]]?
|type="()"}
- <math>f_1' \cdot f_2' \cdot f_3'</math>
+ <math>f_1' \cdot f_2 \cdot f_3 + f_1 \cdot f_2' \cdot f_3 + f_1 \cdot f_2 \cdot f_3'</math>
|| See [[product rule for differentiation#Statement for multiple functions]]
- <math>f_1 \cdot f_2' \cdot f_3' + f_1' \cdot f_2 \cdot f_3' + f_1' \cdot f_2' \cdot f_3</math>
- <math>f_1' \cdot f_2 + f_2' \cdot f_3 + f_3' \cdot f_1</math>
- <math>f_1'' \cdot f_2' \cdot f_3</math>
</quiz>

==Significance==

===Qualitative and existential significance===

Corresponds to [[Product rule for differentiation#Qualitative and existential significance]].

General difficulty level of questions in this section: College level (unless otherwise specified).

<quiz display=simple>
{Suppose <math>f</math> and <math>g</math> are continuous functions at <math>x_0</math> and <math>f \cdot g</math> is the [[pointwise product of functions]]. Which of the following is ''true'' (see last two options!)?
|type="()"}
- If <math>f</math> and <math>g</math> are both left differentiable at <math>x_0</math>, then so is <math>f \cdot g</math>.
- If <math>f</math> and <math>g</math> are both right differentiable at <math>x_0</math>, then so is <math>f \cdot g</math>.
- If <math>f</math> and <math>g</math> are both differentiable at <math>x_0</math>, then so is <math>f \cdot g</math>.
+ All of the above are true
- None of the above is true

{Suppose <math>f</math> and <math>g</math> are continuous functions at <math>x_0</math> and <math>f \cdot g</math> is the [[pointwise product of functions]]. What is the relationship between the differentiability of <math>f</math>, <math>g</math>, and <math>f \cdot g</math> at <math>x_0</math>?
|type="()"}
- If any two of the three functions are differentiable at <math>x_0</math>, then so is the third.
- If <math>f \cdot g</math> is differentiable at <math>x_0</math>, so are <math>f</math> and <math>g</math>.
- If <math>f \cdot g</math> and <math>f</math> are differentiable at <math>x_0</math>, so is <math>g</math>. However, differentiability of <math>f</math> and <math>g</math> at <math>x_0</math> does not guarantee differentiability of <math>f \cdot g</math>.
+ If <math>f</math> and <math>g</math> are both differentiable at <math>x_0</math>, so is <math>f \cdot g</math>. However, differentiability of <math>f \cdot g</math> and <math>f</math> does not guarantee differentiability of <math>g</math>, and differentiability of <math>f \cdot g</math> and <math>g</math> does not guarantee differentiability of <math>f</math>.
- We cannot draw any inferences about differentiability of one of the three functions based on differentiability of the other two.

{Suppose <math>f</math> and <math>g</math> are continuous functions defined on all of <math>\R</math>. Suppose <math>A</math> is the subset of <math>\R</math> comprising those points where <math>f</math> is differentiable, and <matH>B</math> is the subset of <math>\R</math> comprising those points where <math>g</math> is differentiable. Then, what can we say is '''definitely true''' about the subset of <math>\R</math> comprising those points where the [[pointwise product of functions]] <math>f \cdot g</math> is differentiable?
|type="()"}
- It is contained in the intersection <math>A \cap B</math>
+ It contains the intersection <math>A \cap B</math>
- It is contained in the union <math>A \cup B</math>
- It contains the union <math>A \cup B</math>
- None of the above

{Suppose <math>f</math> and <math>g</math> are continuous functions defined on all of <math>\R</math>. Suppose <math>a,b \in \R</math> are such that <math>1 < a < b</math>. Suppose <math>f</math> is known to be differentiable at <math>a</math> and <math>g</math> is known to be differentiable at <math>b</math>. We do not have information about where else <math>f</math or <math>g</math> is differentiable. What can we conclude about where the [[pointwise product of functions]] <math>f \cdot g</math> is differentiable?
|type="()"}
- <math>f \cdot g</math> is differentiable at <math>ab</math>
- <math>f \cdot g</math> is differentiable at at least one of <math>a</math> or <math>b</math> but not necessarily at both
- <math>f \cdot g</math> is differentiable at both <math>a</math> and <math>b</math>
+ We don't have enough information to conclude anything about the set of points where <math>f \cdot g</math> is differentiable

{Suppose <math>\mathcal{F}</math> is a collection of differentiable functions defined on all of <math>\R</math>. Further, suppose that there is a collection <math>\mathcal{B}</math> of functions such that every element of <math>\mathcal{F}</math> can be written as a polynomial in terms of the elements of <math>\mathcal{B}</math>, with constant coefficients. Suppose that the derivative of every element of <math>\mathcal{B}</math> is in <math>\mathcal{F}</math>. Which of the following conditions are sufficient to ensure that the derivative of every element of <math>\mathcal{F}</math> is in <math>\mathcal{F}</math>?
|type="()"}
- It is sufficient to ensure that <math>\mathcal{F}</math> is closed under addition and scalar multiplication, i.e., it forms a [[vector space]] of functions.
- It is sufficient to ensure that <math>\mathcal{F}</math> is closed under multiplication, i.e., the product of any two elements of <math>\mathcal{F}</math> is in <math>\mathcal{F}</math>.
+ It is sufficient to ensure that <math>\mathcal{F}</math> is closed under addition-cum-scalar multiplication (i.e., it forms a vector space), ''and'' closed under multiplication, but just having one of those conditions need not suffice.
- It is not sufficient to ensure that <math>\mathcal{F}</math> is closed under addition-cum-scalar multiplication (i.e., it forms a vector space), ''and'' closed under multiplication
</quiz>

===Computational feasibility significance===

See the section [[#Practical]].

===Computational results significance===

Corresponds to [[Product rule for differentiation#Computational results significance]].

General difficulty level of questions in this section: College level (unless otherwise specified).

<quiz display=simple>
{Suppose <math>f</math> and <math>g</math> are infinitely differentiable functions on all of <math>\R</math> such that both <math>f'</math> and <math>g'</math> are [[periodic function]]s with the same period <math>h > 0</math>. What can we conclude about <math>f \cdot g</math>?
|type="()"}
- <math>f \cdot g</math> must be periodic
- <math>(f \cdot g)'</math> must be periodic, but <math>f \cdot g</math> may or may not be periodic.
- <math>(f \cdot g)''</math> must be periodic, but <math>(f \cdot g)'</math> may or may not be periodic.
- <math>(f \cdot g)'''</math> must be periodic, but <math>(f \cdot g)''</math> may or may not be periodic.
+ We cannot conclude from the given information whether <math>f \cdot g</math> or any of the derivatives of <math>f \cdot g</math> is periodic.

{Suppose <math>f</math> and <math>g</math> are functions defined and differentiable on the open interval <math>(0,1)</math>. Suppose, further, that on <math>(0,1)</math>, the derivative functions <math>f'</math> and <math>g'</math> are both expressible as [[rational function]]s. What can we say about <math>f \cdot g</math> and <math>(f \cdot g)'</math> on <math>(0,1)</math>?
|type="()"}
- Both <math>f \cdot g</math> and <math>(f \cdot g)'</math> are expressible as rational functions.
- <math>f \cdot g</math> is expressible as a rational function, but <math>(f \cdot g)'</math> need not be expressible as a rational function.
- <math>(f \cdot g)'</math> is expressible as a rational function, but <math>f \cdot g</math> need not be expressible as a rational function.
+ Neither <math>f \cdot g</math> nor <math>(f \cdot g)'</math> need be expressible as a rational function.
</quiz>

Quiz:Product rule for differentiation

2024-04-11T05:41:47Z

Vipul: /* Qualitative and existential significance */

{{perspectives}}

For a quiz that tests all the differentiation rules together, see [[Quiz:Differentiation rules]].

==Practical==

Corresponds to [[Practical:Product rule for differentiation]].

General difficulty level of questions in this section: School level (unless otherwise specified).
<quiz display=simple>
{Suppose <math>f</math> and <math>g</math> are both defined and differentiable at the point 1. Suppose <math>\! f(1) = 2, g(1) = 5, f'(1) = 4, g'(1) = 11</math>. What is the value of <math>(f \cdot g)'(1)</math> where <math>f \cdot g</math> denotes the [[pointwise product of functions]]?
|type="()"}
+ 42
- 44
- 54
- 63
- The information given is insufficient to find <math>(f \cdot g)'(1)</math>.

{What is the derivative of the function <math>x \mapsto \exp(x) \sin x</math>? Hint for derivatives of individual functions: <toggledisplay>Derivative of exponential function is exponential function, derivative of sine function is cosine function.</toggledisplay>
|type="()"}
+ <math>x \mapsto \exp(x)(\sin x + \cos x)</math>
- <math>x \mapsto \exp(x)(\cos x - \sin x)</math>
- <math>x \mapsto \exp(x)(\sin x - \cos x)</math>
- <math>x \mapsto \exp(x)\cos x + \exp(1) \sin x</math>
- <math>x \mapsto \exp(x)\sin x + \exp(1) \cos x</math>

{What is the derivative of the function <math>x \mapsto \sqrt{x}\sin(x^2)</math> for <math>x > 0</math>? This question also requires use of [[chain rule for differentiation]].
|type="()"}
- <math>x \mapsto \cos(x^2)/(2\sqrt{x})</math>
- <math>x \mapsto 2\sqrt{x}\cos(x^2)</math>
- <math>x \mapsto 2\sqrt{x}(\cos(x^2 + \sin(x^2))</math>
- <math>x \mapsto 2x\sqrt{x}\sin(x^2) + \cos(x^2)/(2 \sqrt{x})</math>
+ <math>x \mapsto 2x\sqrt{x}\cos(x^2) + \sin(x^2)/(2 \sqrt{x})</math>

{What is the derivative of the function <math>x \mapsto x \sin x \ln x</math> for <math>x > 0</math>? Hint for derivatives of individual functions: <toggledisplay>Derivative of <math>\sin</math> is <math>\cos</math>, derivative of <math>\ln</math> is <math>x \mapsto 1/x</math></toggledisplay>
|type="()"}
- <math>x \mapsto (\cos x)/x</math>
- <math>x \mapsto (-\cos x)/x</math>
- <math>x \mapsto \cos x \ln x + \cos x + (\sin x)/x</math>
- <math>x \mapsto \cos x \ln x - \cos x + (\sin x)/x</math>
+ <math>x \mapsto \sin x \ln x + x \cos x \ln x + \sin x</math>

</quiz>

==Formulas==

General difficulty level of questions in this section: College level (unless otherwise specified)

<quiz display=simple>
{Suppose <math>f</math> and <math>g</math> are both twice differentiable functions everywhere on <math>\R</math>. Which of the following is the correct formula for <math>(f \cdot g)''</math>, the second derivative of the [[pointwise product of functions]]?
|type="()"}
- <math>f'' \cdot g + f \cdot g''</math>
- <math>f'' \cdot g + f' \cdot g' + f \cdot g''</math>
+ <math>f'' \cdot g + 2f' \cdot g' + f \cdot g''</math>
|| See [[product rule for differentiation]], [[product rule for higher derivatives]].
- <math>f'' \cdot g - f' \cdot g' + f \cdot g''</math>
- <math>f'' \cdot g - 2f' \cdot g' + f \cdot g''</math>

{ Suppose <math>f_1,f_2,f_3</math> are everywhere differentiable functions from <math>\R</math> to <math>\R</math>. What is the derivative <math>(f_1 \cdot f_2 \cdot f_3)'</math>, where <math>f_1 \cdot f_2 \cdot f_3</math> denotes the [[pointwise product of functions]]?
|type="()"}
- <math>f_1' \cdot f_2' \cdot f_3'</math>
+ <math>f_1' \cdot f_2 \cdot f_3 + f_1 \cdot f_2' \cdot f_3 + f_1 \cdot f_2 \cdot f_3'</math>
|| See [[product rule for differentiation#Statement for multiple functions]]
- <math>f_1 \cdot f_2' \cdot f_3' + f_1' \cdot f_2 \cdot f_3' + f_1' \cdot f_2' \cdot f_3</math>
- <math>f_1' \cdot f_2 + f_2' \cdot f_3 + f_3' \cdot f_1</math>
- <math>f_1'' \cdot f_2' \cdot f_3</math>
</quiz>

==Significance==

===Qualitative and existential significance===

Corresponds to [[Product rule for differentiation#Qualitative and existential significance]].

General difficulty level of questions in this section: College level (unless otherwise specified).

<quiz display=simple>
{Suppose <math>f</math> and <math>g</math> are continuous functions at <math>x_0</math> and <math>f \cdot g</math> is the [[pointwise product of functions]]. Which of the following is ''true'' (see last two options!)?
|type="()"}
- If <math>f</math> and <math>g</math> are both left differentiable at <math>x_0</math>, then so is <math>f \cdot g</math>.
- If <math>f</math> and <math>g</math> are both right differentiable at <math>x_0</math>, then so is <math>f \cdot g</math>.
- If <math>f</math> and <math>g</math> are both differentiable at <math>x_0</math>, then so is <math>f \cdot g</math>.
+ All of the above are true
- None of the above is true

{Suppose <math>f</math> and <math>g</math> are continuous functions at <math>x_0</math> and <math>f \cdot g</math> is the [[pointwise product of functions]]. What is the relationship between the differentiability of <math>f</math>, <math>g</math>, and <math>f \cdot g</math> at <math>x_0</math>?
|type="()"}
- If any two of the three functions are differentiable at <math>x_0</math>, then so is the third.
- If <math>f \cdot g</math> is differentiable at <math>x_0</math>, so are <math>f</math> and <math>g</math>.
- If <math>f \cdot g</math> and <math>f</math> are differentiable at <math>x_0</math>, so is <math>g</math>. However, differentiability of <math>f</math> and <math>g</math> at <math>x_0</math> does not guarantee differentiability of <math>f \cdot g</math>.
+ If <math>f</math> and <math>g</math> are both differentiable at <math>x_0</math>, so is <math>f \cdot g</math>. However, differentiability of <math>f \cdot g</math> and <math>f</math> does not guarantee differentiability of <math>g</math>, and differentiability of <math>f \cdot g</math> and <math>g</math> does not guarantee differentiability of <math>f</math>.
- We cannot draw any inferences about differentiability of one of the three functions based on differentiability of the other two.

{Suppose <math>f</math> and <math>g</math> are continuous functions defined on all of <math>\R</math>. Suppose <math>A</math> is the subset of <math>\R</math> comprising those points where <math>f</math> is differentiable, and <matH>B</math> is the subset of <math>\R</math> comprising those points where <math>g</math> is differentiable. Then, what can we say is '''definitely true''' about the subset of <math>\R</math> comprising those points where the [[pointwise product of functions]] <math>f \cdot g</math> is differentiable?
|type="()"}
- It is contained in the intersection <math>A \cap B</math>
+ It contains the intersection <math>A \cap B</math>
- It is contained in the union <math>A \cup B</math>
- It contains the union <math>A \cup B</math>
- None of the above

{Suppose <math>f</math> and <math>g</math> are continuous functions defined on all of <math>\R</math>. Suppose <math>a,b \in \R</math> are such that <math>1 < a < b</math>. Suppose <math>f</math> is known to be differentiable at <math>a</math> and <math>g</math> is known to be differentiable at <math>b</math>. We do not have information about where else <math>f</math or <math>g</math> is differentiable. What can we conclude about where the [[pointwise product of functions]] <math>f \cdot g</math> is differentiable?
|type="()"}
- <math>f \cdot g</math> is differentiable at <math>ab</math>
- <math>f \cdot g</math> is differentiable at at least one of <math>a</math or <math>b</math> but not necessarily at both
- <math>f \cdot g</math> is differentiable at both <math>a</math> and <math>b</math>
+ We don't have enough information to conclude anything about the set of points where <math>f \cdot g</math> is differentiable

{Suppose <math>\mathcal{F}</math> is a collection of differentiable functions defined on all of <math>\R</math>. Further, suppose that there is a collection <math>\mathcal{B}</math> of functions such that every element of <math>\mathcal{F}</math> can be written as a polynomial in terms of the elements of <math>\mathcal{B}</math>, with constant coefficients. Suppose that the derivative of every element of <math>\mathcal{B}</math> is in <math>\mathcal{F}</math>. Which of the following conditions are sufficient to ensure that the derivative of every element of <math>\mathcal{F}</math> is in <math>\mathcal{F}</math>?
|type="()"}
- It is sufficient to ensure that <math>\mathcal{F}</math> is closed under addition and scalar multiplication, i.e., it forms a [[vector space]] of functions.
- It is sufficient to ensure that <math>\mathcal{F}</math> is closed under multiplication, i.e., the product of any two elements of <math>\mathcal{F}</math> is in <math>\mathcal{F}</math>.
+ It is sufficient to ensure that <math>\mathcal{F}</math> is closed under addition-cum-scalar multiplication (i.e., it forms a vector space), ''and'' closed under multiplication, but just having one of those conditions need not suffice.
- It is not sufficient to ensure that <math>\mathcal{F}</math> is closed under addition-cum-scalar multiplication (i.e., it forms a vector space), ''and'' closed under multiplication
</quiz>

===Computational feasibility significance===

See the section [[#Practical]].

===Computational results significance===

Corresponds to [[Product rule for differentiation#Computational results significance]].

General difficulty level of questions in this section: College level (unless otherwise specified).

<quiz display=simple>
{Suppose <math>f</math> and <math>g</math> are infinitely differentiable functions on all of <math>\R</math> such that both <math>f'</math> and <math>g'</math> are [[periodic function]]s with the same period <math>h > 0</math>. What can we conclude about <math>f \cdot g</math>?
|type="()"}
- <math>f \cdot g</math> must be periodic
- <math>(f \cdot g)'</math> must be periodic, but <math>f \cdot g</math> may or may not be periodic.
- <math>(f \cdot g)''</math> must be periodic, but <math>(f \cdot g)'</math> may or may not be periodic.
- <math>(f \cdot g)'''</math> must be periodic, but <math>(f \cdot g)''</math> may or may not be periodic.
+ We cannot conclude from the given information whether <math>f \cdot g</math> or any of the derivatives of <math>f \cdot g</math> is periodic.

{Suppose <math>f</math> and <math>g</math> are functions defined and differentiable on the open interval <math>(0,1)</math>. Suppose, further, that on <math>(0,1)</math>, the derivative functions <math>f'</math> and <math>g'</math> are both expressible as [[rational function]]s. What can we say about <math>f \cdot g</math> and <math>(f \cdot g)'</math> on <math>(0,1)</math>?
|type="()"}
- Both <math>f \cdot g</math> and <math>(f \cdot g)'</math> are expressible as rational functions.
- <math>f \cdot g</math> is expressible as a rational function, but <math>(f \cdot g)'</math> need not be expressible as a rational function.
- <math>(f \cdot g)'</math> is expressible as a rational function, but <math>f \cdot g</math> need not be expressible as a rational function.
+ Neither <math>f \cdot g</math> nor <math>(f \cdot g)'</math> need be expressible as a rational function.
</quiz>

Quiz:Product rule for differentiation

2024-04-11T05:30:54Z

Vipul: /* Computational results significance */

{{perspectives}}

For a quiz that tests all the differentiation rules together, see [[Quiz:Differentiation rules]].

==Practical==

Corresponds to [[Practical:Product rule for differentiation]].

General difficulty level of questions in this section: School level (unless otherwise specified).
<quiz display=simple>
{Suppose <math>f</math> and <math>g</math> are both defined and differentiable at the point 1. Suppose <math>\! f(1) = 2, g(1) = 5, f'(1) = 4, g'(1) = 11</math>. What is the value of <math>(f \cdot g)'(1)</math> where <math>f \cdot g</math> denotes the [[pointwise product of functions]]?
|type="()"}
+ 42
- 44
- 54
- 63
- The information given is insufficient to find <math>(f \cdot g)'(1)</math>.

{What is the derivative of the function <math>x \mapsto \exp(x) \sin x</math>? Hint for derivatives of individual functions: <toggledisplay>Derivative of exponential function is exponential function, derivative of sine function is cosine function.</toggledisplay>
|type="()"}
+ <math>x \mapsto \exp(x)(\sin x + \cos x)</math>
- <math>x \mapsto \exp(x)(\cos x - \sin x)</math>
- <math>x \mapsto \exp(x)(\sin x - \cos x)</math>
- <math>x \mapsto \exp(x)\cos x + \exp(1) \sin x</math>
- <math>x \mapsto \exp(x)\sin x + \exp(1) \cos x</math>

{What is the derivative of the function <math>x \mapsto \sqrt{x}\sin(x^2)</math> for <math>x > 0</math>? This question also requires use of [[chain rule for differentiation]].
|type="()"}
- <math>x \mapsto \cos(x^2)/(2\sqrt{x})</math>
- <math>x \mapsto 2\sqrt{x}\cos(x^2)</math>
- <math>x \mapsto 2\sqrt{x}(\cos(x^2 + \sin(x^2))</math>
- <math>x \mapsto 2x\sqrt{x}\sin(x^2) + \cos(x^2)/(2 \sqrt{x})</math>
+ <math>x \mapsto 2x\sqrt{x}\cos(x^2) + \sin(x^2)/(2 \sqrt{x})</math>

{What is the derivative of the function <math>x \mapsto x \sin x \ln x</math> for <math>x > 0</math>? Hint for derivatives of individual functions: <toggledisplay>Derivative of <math>\sin</math> is <math>\cos</math>, derivative of <math>\ln</math> is <math>x \mapsto 1/x</math></toggledisplay>
|type="()"}
- <math>x \mapsto (\cos x)/x</math>
- <math>x \mapsto (-\cos x)/x</math>
- <math>x \mapsto \cos x \ln x + \cos x + (\sin x)/x</math>
- <math>x \mapsto \cos x \ln x - \cos x + (\sin x)/x</math>
+ <math>x \mapsto \sin x \ln x + x \cos x \ln x + \sin x</math>

</quiz>

==Formulas==

General difficulty level of questions in this section: College level (unless otherwise specified)

<quiz display=simple>
{Suppose <math>f</math> and <math>g</math> are both twice differentiable functions everywhere on <math>\R</math>. Which of the following is the correct formula for <math>(f \cdot g)''</math>, the second derivative of the [[pointwise product of functions]]?
|type="()"}
- <math>f'' \cdot g + f \cdot g''</math>
- <math>f'' \cdot g + f' \cdot g' + f \cdot g''</math>
+ <math>f'' \cdot g + 2f' \cdot g' + f \cdot g''</math>
|| See [[product rule for differentiation]], [[product rule for higher derivatives]].
- <math>f'' \cdot g - f' \cdot g' + f \cdot g''</math>
- <math>f'' \cdot g - 2f' \cdot g' + f \cdot g''</math>

{ Suppose <math>f_1,f_2,f_3</math> are everywhere differentiable functions from <math>\R</math> to <math>\R</math>. What is the derivative <math>(f_1 \cdot f_2 \cdot f_3)'</math>, where <math>f_1 \cdot f_2 \cdot f_3</math> denotes the [[pointwise product of functions]]?
|type="()"}
- <math>f_1' \cdot f_2' \cdot f_3'</math>
+ <math>f_1' \cdot f_2 \cdot f_3 + f_1 \cdot f_2' \cdot f_3 + f_1 \cdot f_2 \cdot f_3'</math>
|| See [[product rule for differentiation#Statement for multiple functions]]
- <math>f_1 \cdot f_2' \cdot f_3' + f_1' \cdot f_2 \cdot f_3' + f_1' \cdot f_2' \cdot f_3</math>
- <math>f_1' \cdot f_2 + f_2' \cdot f_3 + f_3' \cdot f_1</math>
- <math>f_1'' \cdot f_2' \cdot f_3</math>
</quiz>

==Significance==

===Qualitative and existential significance===

Corresponds to [[Product rule for differentiation#Qualitative and existential significance]].

General difficulty level of questions in this section: College level (unless otherwise specified).

<quiz display=simple>
{Suppose <math>f</math> and <math>g</math> are continuous functions at <math>x_0</math> and <math>f \cdot g</math> is the [[pointwise product of functions]]. Which of the following is ''true'' (see last two options!)?
|type="()"}
- If <math>f</math> and <math>g</math> are both left differentiable at <math>x_0</math>, then so is <math>f \cdot g</math>.
- If <math>f</math> and <math>g</math> are both right differentiable at <math>x_0</math>, then so is <math>f \cdot g</math>.
- If <math>f</math> and <math>g</math> are both differentiable at <math>x_0</math>, then so is <math>f \cdot g</math>.
+ All of the above are true
- None of the above is true

{Suppose <math>f</math> and <math>g</math> are continuous functions at <math>x_0</math> and <math>f \cdot g</math> is the [[pointwise product of functions]]. What is the relationship between the differentiability of <math>f</math>, <math>g</math>, and <math>f \cdot g</math> at <math>x_0</math>?
|type="()"}
- If any two of the three functions are differentiable at <math>x_0</math>, then so is the third.
- If <math>f \cdot g</math> is differentiable at <math>x_0</math>, so are <math>f</math> and <math>g</math>.
- If <math>f \cdot g</math> and <math>f</math> are differentiable at <math>x_0</math>, so is <math>g</math>. However, differentiability of <math>f</math> and <math>g</math> at <math>x_0</math> does not guarantee differentiability of <math>f \cdot g</math>.
+ If <math>f</math> and <math>g</math> are both differentiable at <math>x_0</math>, so is <math>f \cdot g</math>. However, differentiability of <math>f \cdot g</math> and <math>f</math> does not guarantee differentiability of <math>g</math>, and differentiability of <math>f \cdot g</math> and <math>g</math> does not guarantee differentiability of <math>f</math>.
- We cannot draw any inferences about differentiability of one of the three functions based on differentiability of the other two.

{Suppose <math>f</math> and <math>g</math> are continuous functions defined on all of <math>\R</math>. Suppose <math>A</math> is the subset of <math>\R</math> comprising those points where <math>f</math> is differentiable, and <matH>B</math> is the subset of <math>\R</math> comprising those points where <math>g</math> is differentiable. Then, what can we say is '''definitely true''' about the subset of <math>\R</math> comprising those points where the [[pointwise product of functions]] <math>f \cdot g</math> is differentiable?
|type="()"}
- It is contained in the intersection <math>A \cap B</math>
+ It contains the intersection <math>A \cap B</math>
- It is contained in the union <math>A \cup B</math>
- It contains the union <math>A \cup B</math>
- None of the above

{Suppose <math>\mathcal{F}</math> is a collection of differentiable functions defined on all of <math>\R</math>. Further, suppose that there is a collection <math>\mathcal{B}</math> of functions such that every element of <math>\mathcal{F}</math> can be written as a polynomial in terms of the elements of <math>\mathcal{B}</math>, with constant coefficients. Suppose that the derivative of every element of <math>\mathcal{B}</math> is in <math>\mathcal{F}</math>. Which of the following conditions are sufficient to ensure that the derivative of every element of <math>\mathcal{F}</math> is in <math>\mathcal{F}</math>?
|type="()"}
- It is sufficient to ensure that <math>\mathcal{F}</math> is closed under addition and scalar multiplication, i.e., it forms a [[vector space]] of functions.
- It is sufficient to ensure that <math>\mathcal{F}</math> is closed under multiplication, i.e., the product of any two elements of <math>\mathcal{F}</math> is in <math>\mathcal{F}</math>.
+ It is sufficient to ensure that <math>\mathcal{F}</math> is closed under addition-cum-scalar multiplication (i.e., it forms a vector space), ''and'' closed under multiplication, but just having one of those conditions need not suffice.
- It is not sufficient to ensure that <math>\mathcal{F}</math> is closed under addition-cum-scalar multiplication (i.e., it forms a vector space), ''and'' closed under multiplication
</quiz>

===Computational feasibility significance===

See the section [[#Practical]].

===Computational results significance===

Corresponds to [[Product rule for differentiation#Computational results significance]].

General difficulty level of questions in this section: College level (unless otherwise specified).

<quiz display=simple>
{Suppose <math>f</math> and <math>g</math> are infinitely differentiable functions on all of <math>\R</math> such that both <math>f'</math> and <math>g'</math> are [[periodic function]]s with the same period <math>h > 0</math>. What can we conclude about <math>f \cdot g</math>?
|type="()"}
- <math>f \cdot g</math> must be periodic
- <math>(f \cdot g)'</math> must be periodic, but <math>f \cdot g</math> may or may not be periodic.
- <math>(f \cdot g)''</math> must be periodic, but <math>(f \cdot g)'</math> may or may not be periodic.
- <math>(f \cdot g)'''</math> must be periodic, but <math>(f \cdot g)''</math> may or may not be periodic.
+ We cannot conclude from the given information whether <math>f \cdot g</math> or any of the derivatives of <math>f \cdot g</math> is periodic.

{Suppose <math>f</math> and <math>g</math> are functions defined and differentiable on the open interval <math>(0,1)</math>. Suppose, further, that on <math>(0,1)</math>, the derivative functions <math>f'</math> and <math>g'</math> are both expressible as [[rational function]]s. What can we say about <math>f \cdot g</math> and <math>(f \cdot g)'</math> on <math>(0,1)</math>?
|type="()"}
- Both <math>f \cdot g</math> and <math>(f \cdot g)'</math> are expressible as rational functions.
- <math>f \cdot g</math> is expressible as a rational function, but <math>(f \cdot g)'</math> need not be expressible as a rational function.
- <math>(f \cdot g)'</math> is expressible as a rational function, but <math>f \cdot g</math> need not be expressible as a rational function.
+ Neither <math>f \cdot g</math> nor <math>(f \cdot g)'</math> need be expressible as a rational function.
</quiz>

Quiz:Product rule for differentiation

2024-04-11T05:29:41Z

Vipul: /* Qualitative and existential significance */

{{perspectives}}

For a quiz that tests all the differentiation rules together, see [[Quiz:Differentiation rules]].

==Practical==

Corresponds to [[Practical:Product rule for differentiation]].

General difficulty level of questions in this section: School level (unless otherwise specified).
<quiz display=simple>
{Suppose <math>f</math> and <math>g</math> are both defined and differentiable at the point 1. Suppose <math>\! f(1) = 2, g(1) = 5, f'(1) = 4, g'(1) = 11</math>. What is the value of <math>(f \cdot g)'(1)</math> where <math>f \cdot g</math> denotes the [[pointwise product of functions]]?
|type="()"}
+ 42
- 44
- 54
- 63
- The information given is insufficient to find <math>(f \cdot g)'(1)</math>.

{What is the derivative of the function <math>x \mapsto \exp(x) \sin x</math>? Hint for derivatives of individual functions: <toggledisplay>Derivative of exponential function is exponential function, derivative of sine function is cosine function.</toggledisplay>
|type="()"}
+ <math>x \mapsto \exp(x)(\sin x + \cos x)</math>
- <math>x \mapsto \exp(x)(\cos x - \sin x)</math>
- <math>x \mapsto \exp(x)(\sin x - \cos x)</math>
- <math>x \mapsto \exp(x)\cos x + \exp(1) \sin x</math>
- <math>x \mapsto \exp(x)\sin x + \exp(1) \cos x</math>

{What is the derivative of the function <math>x \mapsto \sqrt{x}\sin(x^2)</math> for <math>x > 0</math>? This question also requires use of [[chain rule for differentiation]].
|type="()"}
- <math>x \mapsto \cos(x^2)/(2\sqrt{x})</math>
- <math>x \mapsto 2\sqrt{x}\cos(x^2)</math>
- <math>x \mapsto 2\sqrt{x}(\cos(x^2 + \sin(x^2))</math>
- <math>x \mapsto 2x\sqrt{x}\sin(x^2) + \cos(x^2)/(2 \sqrt{x})</math>
+ <math>x \mapsto 2x\sqrt{x}\cos(x^2) + \sin(x^2)/(2 \sqrt{x})</math>

{What is the derivative of the function <math>x \mapsto x \sin x \ln x</math> for <math>x > 0</math>? Hint for derivatives of individual functions: <toggledisplay>Derivative of <math>\sin</math> is <math>\cos</math>, derivative of <math>\ln</math> is <math>x \mapsto 1/x</math></toggledisplay>
|type="()"}
- <math>x \mapsto (\cos x)/x</math>
- <math>x \mapsto (-\cos x)/x</math>
- <math>x \mapsto \cos x \ln x + \cos x + (\sin x)/x</math>
- <math>x \mapsto \cos x \ln x - \cos x + (\sin x)/x</math>
+ <math>x \mapsto \sin x \ln x + x \cos x \ln x + \sin x</math>

</quiz>

==Formulas==

General difficulty level of questions in this section: College level (unless otherwise specified)

<quiz display=simple>
{Suppose <math>f</math> and <math>g</math> are both twice differentiable functions everywhere on <math>\R</math>. Which of the following is the correct formula for <math>(f \cdot g)''</math>, the second derivative of the [[pointwise product of functions]]?
|type="()"}
- <math>f'' \cdot g + f \cdot g''</math>
- <math>f'' \cdot g + f' \cdot g' + f \cdot g''</math>
+ <math>f'' \cdot g + 2f' \cdot g' + f \cdot g''</math>
|| See [[product rule for differentiation]], [[product rule for higher derivatives]].
- <math>f'' \cdot g - f' \cdot g' + f \cdot g''</math>
- <math>f'' \cdot g - 2f' \cdot g' + f \cdot g''</math>

{ Suppose <math>f_1,f_2,f_3</math> are everywhere differentiable functions from <math>\R</math> to <math>\R</math>. What is the derivative <math>(f_1 \cdot f_2 \cdot f_3)'</math>, where <math>f_1 \cdot f_2 \cdot f_3</math> denotes the [[pointwise product of functions]]?
|type="()"}
- <math>f_1' \cdot f_2' \cdot f_3'</math>
+ <math>f_1' \cdot f_2 \cdot f_3 + f_1 \cdot f_2' \cdot f_3 + f_1 \cdot f_2 \cdot f_3'</math>
|| See [[product rule for differentiation#Statement for multiple functions]]
- <math>f_1 \cdot f_2' \cdot f_3' + f_1' \cdot f_2 \cdot f_3' + f_1' \cdot f_2' \cdot f_3</math>
- <math>f_1' \cdot f_2 + f_2' \cdot f_3 + f_3' \cdot f_1</math>
- <math>f_1'' \cdot f_2' \cdot f_3</math>
</quiz>

==Significance==

===Qualitative and existential significance===

Corresponds to [[Product rule for differentiation#Qualitative and existential significance]].

General difficulty level of questions in this section: College level (unless otherwise specified).

<quiz display=simple>
{Suppose <math>f</math> and <math>g</math> are continuous functions at <math>x_0</math> and <math>f \cdot g</math> is the [[pointwise product of functions]]. Which of the following is ''true'' (see last two options!)?
|type="()"}
- If <math>f</math> and <math>g</math> are both left differentiable at <math>x_0</math>, then so is <math>f \cdot g</math>.
- If <math>f</math> and <math>g</math> are both right differentiable at <math>x_0</math>, then so is <math>f \cdot g</math>.
- If <math>f</math> and <math>g</math> are both differentiable at <math>x_0</math>, then so is <math>f \cdot g</math>.
+ All of the above are true
- None of the above is true

{Suppose <math>f</math> and <math>g</math> are continuous functions at <math>x_0</math> and <math>f \cdot g</math> is the [[pointwise product of functions]]. What is the relationship between the differentiability of <math>f</math>, <math>g</math>, and <math>f \cdot g</math> at <math>x_0</math>?
|type="()"}
- If any two of the three functions are differentiable at <math>x_0</math>, then so is the third.
- If <math>f \cdot g</math> is differentiable at <math>x_0</math>, so are <math>f</math> and <math>g</math>.
- If <math>f \cdot g</math> and <math>f</math> are differentiable at <math>x_0</math>, so is <math>g</math>. However, differentiability of <math>f</math> and <math>g</math> at <math>x_0</math> does not guarantee differentiability of <math>f \cdot g</math>.
+ If <math>f</math> and <math>g</math> are both differentiable at <math>x_0</math>, so is <math>f \cdot g</math>. However, differentiability of <math>f \cdot g</math> and <math>f</math> does not guarantee differentiability of <math>g</math>, and differentiability of <math>f \cdot g</math> and <math>g</math> does not guarantee differentiability of <math>f</math>.
- We cannot draw any inferences about differentiability of one of the three functions based on differentiability of the other two.

{Suppose <math>f</math> and <math>g</math> are continuous functions defined on all of <math>\R</math>. Suppose <math>A</math> is the subset of <math>\R</math> comprising those points where <math>f</math> is differentiable, and <matH>B</math> is the subset of <math>\R</math> comprising those points where <math>g</math> is differentiable. Then, what can we say is '''definitely true''' about the subset of <math>\R</math> comprising those points where the [[pointwise product of functions]] <math>f \cdot g</math> is differentiable?
|type="()"}
- It is contained in the intersection <math>A \cap B</math>
+ It contains the intersection <math>A \cap B</math>
- It is contained in the union <math>A \cup B</math>
- It contains the union <math>A \cup B</math>
- None of the above

{Suppose <math>\mathcal{F}</math> is a collection of differentiable functions defined on all of <math>\R</math>. Further, suppose that there is a collection <math>\mathcal{B}</math> of functions such that every element of <math>\mathcal{F}</math> can be written as a polynomial in terms of the elements of <math>\mathcal{B}</math>, with constant coefficients. Suppose that the derivative of every element of <math>\mathcal{B}</math> is in <math>\mathcal{F}</math>. Which of the following conditions are sufficient to ensure that the derivative of every element of <math>\mathcal{F}</math> is in <math>\mathcal{F}</math>?
|type="()"}
- It is sufficient to ensure that <math>\mathcal{F}</math> is closed under addition and scalar multiplication, i.e., it forms a [[vector space]] of functions.
- It is sufficient to ensure that <math>\mathcal{F}</math> is closed under multiplication, i.e., the product of any two elements of <math>\mathcal{F}</math> is in <math>\mathcal{F}</math>.
+ It is sufficient to ensure that <math>\mathcal{F}</math> is closed under addition-cum-scalar multiplication (i.e., it forms a vector space), ''and'' closed under multiplication, but just having one of those conditions need not suffice.
- It is not sufficient to ensure that <math>\mathcal{F}</math> is closed under addition-cum-scalar multiplication (i.e., it forms a vector space), ''and'' closed under multiplication
</quiz>

===Computational feasibility significance===

See the section [[#Practical]].

===Computational results significance===

Corresponds to [[Product rule for differentiation#Computational results significance]].

General difficulty level of questions in this section: College level (unless otherwise specified).

<quiz display=simple>
{Suppose <math>f</math> and <math>g</math> are infinitely differentiable functions on all of <math>\R</math> such that both <math>f'</math> and <math>g'</math> are [[periodic function]]s with the same period <math>h > 0</math>. What can we conclude about <math>f \cdot g</math>?
|type="()"}
- <math>f \cdot g</math> must be periodic
- <math>(f \cdot g)'</math> must be periodic, but <math>f \cdot g</math> may or may not be periodic.
- <math>(f \cdot g)''</math> must be periodic, but <math>(f \cdot g)'</math> may or may not be periodic.
- <math>(f \cdot g)'''</math> must be periodic, but <math>(f \cdot g)''</math> may or may not be periodic.
+ We cannot conclude from the given information whether <math>f \cdot g</math> or any of the derivatives of <math>f \cdot g</math> is periodic.

{Suppose <math>f</math> and <math>g</math> are functions defined and differentiable on the open interval <math>(0,1)</math>. Suppose, further, that on <math>(0,1)</math>, the derivative functions <math>\! f'</math> and <math>\! g'</math> are both expressible as [[rational function]]s. What can we say about <math>f \cdot g</math> and <math>(f \cdot g)'</math> on <math>(0,1)</math>?
|type="()"}
- Both <math>f \cdot g</math> and <math>(f \cdot g)'</math> are expressible as rational functions.
- <math>f \cdot g</math> is expressible as a rational function, but <math>(f \cdot g)'</math> need not be expressible as a rational function.
- <math>(f \cdot g)'</math> is expressible as a rational function, but <math>f \cdot g</math> need not be expressible as a rational function.
+ Neither <math>f \cdot g</math> nor <math>(f \cdot g)'</math> need be expressible as a rational function.
</quiz>

Derivative

2024-04-11T02:39:34Z

Vipul: /* Algebraic definition elaborated in terms of epsilon-delta definition of limits */

{{core term}}

==Name==

The term '''derivative''' is used for the notion defined here. However, there are many variations of the concept of derivative that are described by using adjectives to modify the noun. When these variations are being talked about, it is helpful to provide a similar adjective to indicate that we are talking about the usual notion of derivative. The variations and corresponding terminological clarification are below:

{| class="sortable" border="1"
! Variation of notion of derivative !! Modified name for the usual notion of derivative to emphasize it's the original notion and not the variation
|-
| one-sided derivative (left hand derivative and right hand derivative) -- defined on this page || two-sided derivative
|-
| [[higher derivative]] (obtained by repeated differentiation) || first derivative
|-
| [[partial derivative]] (derivative of a function of multiple variables with respect to one of the variables holding the others constant) || ordinary derivative
|-
| [[discrete derivative]] (not commonly used) || continuous derivative (not commonly used)
|}

==Definition at a point==

===Conceptual definition===

Suppose <math>f</math> is a [[function]] defined on a subset of the reals and <math>x_0</math> is a point in the [[interior]] of the [[domain]] of <math>f</math>, i.e., the domain of <math>f</math> contains an [[open interval]] surrounding <math>x_0</math>. The '''derivative''' of <math>f</math> at <math>x_0</math>, denoted <math>\! f'(x_0)</math>, is the '''instantaneous rate of change''' of <math>f(x)</math> with respect to <math>x</math> at <math>x_0</math>. It is defined as the [[limit]] of the '''average rate of change''' of <math>f</math> between <math>x</math> and <math>x_0</math>, as <math>x</math> approaches <math>x_0</math>.

In the more formal definitions below, we will see that:

* [[Difference quotient]] formalizes the notion of average rate of change.
* Derivative formalizes the notion of instantaneous rate of change, and is the limit of the difference quotient.

===Algebraic definition===

Suppose <math>f</math> is a [[function]] defined on a subset of the reals and <math>x_0</math> is a point in the [[interior]] of the [[domain]] of <math>f</math>, i.e., the domain of <math>f</math> contains an [[open interval]] surrounding <math>x_0</math>. The '''derivative''' (also called '''first derivative''') of <math>f</math> at <math>x_0</math>, denoted <math>f'(x_0)</math> is defined as the [[limit]] of the [[defining ingredient::difference quotient]] of <math>f</math> between <math>x_0</math> and <math>x</math>, as <math>x \to x_0</math>. Explicitly:

<math>\! f'(x_0) := \lim_{x \to x_0} \Delta f(x,x_0) = \lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0}</math>

If this limit exists, then we say that the derivative exists and has this value, and we say that the function is '''differentiable''' at the point. If the limit does not exist, then we say that the function is ''not'' differentiable at the point and the derivative does ''not'' exist.

===Computationally useful version of algebraic definition===

This is obtained from the previous definition by the variable substitution <math>h := x - x_0</math> so <math>x = x_0 + h</math>. Explicitly:

<math>\! f'(x_0) := \lim_{h \to 0} \Delta f(x_0 + h,x_0) = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}</math>

===Geometric definition===

Suppose <math>f</math> is a [[function]] and <math>x_0</math> is a point in the [[interior]] of the [[domain]] of <math>f</math>, i.e., the domain of <math>f</math> contains an [[open interval]] surrounding <math>x_0</math>. The '''derivative''' of <math>f</math> at <math>x_0</math> is the [[slope]] of the [[tangent line]] to the [[graph]] of <math>f</math> through the point <math>(x_0,f(x_0))</math>.

===Algebraic definition elaborated in terms of epsilon-delta definition of limits===

Suppose <math>f</math> is a [[function]] defined on a subset of the reals and <math>x_0</math> is a point in the [[interior]] of the [[domain]] of <math>f</math>, i.e., the domain of <math>f</math> contains an [[open interval]] surrounding <math>x_0</math>. The '''derivative''' (also called '''first derivative''') of <math>f</math> at <math>x_0</math>, denoted <math>f'(x_0)</math>, is defined as a real number <math>L</math> such that:

{{quotation|For every <math>\varepsilon > 0</math> there exists <math>\delta > 0</math> such that if <math>\! 0 < |x - x_0| < \delta</math> then <math>|f(x) - f(x_0) - L(x - x_0)| < \varepsilon|x - x_0|</math>.}}

<center>{{#widget:YouTube|id=cZVLsvvBDgo}}</center>

==Definition as a function==

Suppose <math>f</math> is a [[function]] defined on a subset of the reals. Its '''derivative''' or '''first derivative''', denoted <math>f'</math>, is a function defined as follows:

* The domain is the following subset of the domain of <math>f</math>: An element in the domain of <math>f</math> is in the domain of <math>f'</math> if and only if it is in the ''interior'' of the domain of <math>f</math> and the derivative of <math>f</math> exists at the point.
* The function value at any point in the domain is simply the value of the derivative of <math>f</math> at that point.

{{generic point specific point confusion}}

==One-sided notions==

===Left-hand derivative===

Suppose <math>f</math> is a function defined at a point <math>x_0 \in \R</math> and also to the immediate left of <math>x_0</math>. The '''left-hand derivative''' of <math>f</math> at <math>x_0</math> is defined as the ''left hand limit'' for the difference quotient between <math>x</math> and <math>x_0</math>. In other words, it is:

<math>\operatorname{LHD}(f)(x_0) = f'_-(x_0) := \lim_{x \to x_0^-}\Delta f(x,x_0) = \lim_{x \to x_0^-} \frac{f(x) - f(x_0)}{x - x_0} = \lim_{h \to 0^-} \frac{f(x_0 + h) - f(x_0))}{h}</math>

===Right-hand derivative===

Suppose <math>f</math> is a function defined at a point <math>x_0 \in \R</math> and also to the immediate right of <math>x_0</math>. The '''right-hand derivative''' of <math>f</math> at <math>x_0</math> is defined as the ''right hand limit'' for the difference quotient between <math>x</math> and <math>x_0</math>. In other words, it is:

<math>\operatorname{RHD}(f)(x_0) = f'_+(x_0) := \lim_{x \to x_0^+} \Delta f(x,x_0) = \lim_{x \to x_0^+} \frac{f(x) - f(x_0)}{x - x_0} = \lim_{h \to 0^+} \frac{f(x_0 + h) - f(x_0)}{h}</math>

===Relation between one-sided derivatives and the usual (two-sided) derivative===

The derivative <math>f'(x_0)</math> exists if and only if (both the left hand derivative and the right hand derivative exist at <math>x_0</math> and their values are equal). Further, the value of the derivative equals both these equals values.

==Leibniz notation for derivative==

The Leibniz notation for derivative views the derivative as the ''relative rate of change of two variables'' and is thus a somewhat different perspective on the derivative.

Suppose <math>f</math> is a function, and <math>x,y</math> are variables related by <math>y := f(x)</math>. Here, <math>x</math> is an ''independent variable'' and <math>y</math> is the ''dependent variable'' (with the dependency being described by the function <math>f</math>). We then define:

<math>\frac{dy}{dx} := f'(x)</math>

In particular, <math>dy/dx</math> is a ''function'' of <math>x</math>. Its value at <math>x = x_0</math> is defined as <math>f'(x_0)</math> and is denoted as follows:

<math>\! \frac{dy}{dx} |_{x = x_0} := f'(x_0)</math>

Note that the <math>dy/dx</math> notation does ''not'' mean that a number <math>dy</math> is being divided by a number <math>dx</math>. One way of justifying this notation is by expressing it as a limit of a [[difference quotient]]; here, <math>y_0 = f(x_0)</math>:

<math>\! \frac{dy}{dx} |_{x = x_0} := \lim_{x \to x_0} \frac{y - y_0}{x - x_0} = \lim_{x \to x_0} \frac{\Delta y}{\Delta x}</math>

where <math>\Delta y = y - y_0</math> denotes the ''difference'' in <math>y</math>-values and <math>\Delta x = x - x_0</math> denotes the ''difference'' in <math>x</math>-values.

===Quotient notation is misleading but salvageable===

The difference quotient is actually a quotient of ''numbers'', and the derivative is a limit of this. Hence, many of the formal manipulations involving fractions of numbers work with this notation, even though <math>dy/dx</math> ''itself'' is not a quotient of numbers (see [[chain rule for differentiation]] and [[inverse function theorem]]).

===Expressive advantage of Leibniz notation===

The Leibniz notation is advantageous for carrying out computations by hand and writing derivative expressions because it does not require us to ''name'' every function in order to differentiate it. On the other hand, the prime notation requires us to name a function before we can talk of its derivatives.

Thus, the Leibniz notation is crucial for constructing complicated expressions involving derivatives. For instance, consider the expression:

<math>\frac{d}{dx}\left[\left(\frac{d}{dx}(x - \cos x)\right)\sin^2\left(\frac{d}{dx}(x^2 \cos(x^3))\right)\right]</math>

In order to write this expression with the prime notation, we would first need to give names to functions <math>x - \cos x</math> and <math>x^2 \cos x^3</math>, then give a name to the entire expression within square braces, and then talk of differentiating it.

===Expressive disadvantage of Leibniz notation===

The Leibniz notation is not ''point-free'', i.e., we have to use a symbol to denote the point at which the function is being applied. In contrast, with the prime notation, we can make statements like <math>\! \sin' = \cos</math>.

==Physical applications==

===Note on units===

In applications to the natural and social sciences, the ''units'' used for measuring <math>dy/dx</math> are the units used for measuring <math>y</math> divided by the units used for measuring <math>x</math>. This is because the derivative is a limit of a difference quotient, which is a quantity measured in units of <math>y</math> divided by a quantity measured in units of <math>x</math>.

If the dimensions are expressed using a framework such as the MLT framework for physical quantities, then the MLT exponents subtract.

{| class="sortable" border="1"
! Context !! Example of derivative from real world application !! Functionally dependent variable being differentiated !! Independent variable in terms of which differentiation is happening !! Corresponding difference quotient !! Term for numerator of difference quotient !! Term for denominator of difference quotient !! Comment
|-
| kinematics (classical mechanics, physics) || instantaneous velocity measured in units of length/time || position measured in units of length || time measured in units of time || average velocity || displacement || time elapsed || strictly speaking, this is a vector-valued derivative, but we can use single variable calculus if we restrict to motion along a straight line.
|-
| kinematics (classical mechanics, physics) || instantaneous acceleration measured in units of length/(time)^2 || velocity measured in units of length/time || time measured in units of time || average acceleration || change in velocity || time elapsed || strictly speaking, this is a vector-valued derivative, but we can use single variable calculus if we restrict to motion along a straight line.
|-
| chemical reaction (chemistry) || rate of change of concentration of a particular reaction product measured in units of (concentration measure)/(time). || concentration Suitable concentration measure could be molarity (for reactions in solution) or partial pressure (for gaseous reactions) || time measured in units of time || average rate of change of concentration of the product measured in units of (concentration measure)/(time). || change in concentration of product || time elapsed ||
|}

==Related notions==

{| class="sortable" border="1"
! Notion !! How it relates to derivative
|-
| [[higher derivative]] || differentiate again the function obtained by differentiating a particular function, and apply this process repeatedly. Specifically, the <math>k^{th}</math> derivative is the function obtained by applying the differentiation operation <math>k</math> times.
|-
| [[antiderivative]] || a function that has the given derivative. Antidifferentiation is the reverse of differentiation. The general expression for the antiderivative is also called the ''indefinite integral''.
|-
| [[partial derivative]] || a function of more than one variable is differentiated with respect to one of the variables keeping the others constant.
|-
| [[higher partial derivative]] || obtained by applying the partial differentiation operation to a function of more than one variable. The ''pure'' higher partials are those where all the partial differentiation operations are with respect to the same variable. The ''mixed'' higher partials are those where the partial differentiation operations are with respect to more than one variable.
|-
| [[differential]] || {{fillin}}
|}

==Significance==

===Significance of sign on intervals===

The derivative represents the ''rate of change'', and roughly speaking, the sign of derivative represents the ''direction of change''. We list the loose and precise statements below:

{| class="sortable" border="1"
! Loose statement !! Precise versions
|-
| [[increasing function]] means positive derivative || [[positive derivative implies increasing]] [[nonnegative derivative that is not identically zero on any interval implies increasing]] [[increasing and differentiable implies nonnegative derivative that is not identically zero on any interval]]
|-
| [[decreasing function]] means negative derivative || [[negative derivative implies decreasing]] [[decreasing and differentiable implies nonpositive derivative that is not identically zero on any interval]]
|-
| [[constant function]] means zero derivative || [[constant function implies zero derivative]] [[zero derivative implies locally constant]]
|}

===Significance of sign at points===

This is quite similar to the significance on an interval, but the behavior at individual points can be anomalous and can also represent transitions between different kinds of intervals.

{| class="sortable" border="1"
! Loose statement !! Precise versions
|-
| comparison of function value with points on immediate left/right tells us sign of one-sided derivative and vice versa || [[local maximum from the left implies left hand derivative is nonnegative if it exists]] [[local maximum from the right implies right hand derivative is nonpositive if it exists]] [[local minimum from the left implies left hand derivative is nonpositive if it exists]] [[.ocal minimum from the right implies right hand derivative is nonnegative if it exists]]
|-
| local maximum/minimum value must occur at [[critical point]], which is a point where derivative is zero or does not exist. Moreover, sign of derivative on immediate left and right help determine whether it is local max, min, or neither. || [[first derivative test]], see also [[second derivative test]] and [[higher derivative tests]].
|}

==Computation of derivative==

=== List of most commonly used rules ===

<section begin="differentiation rules"/>

For a full list, see [[:Category:Differentiation rules]].

{| class="sortable" border="1"
! Method for constructing new functions from old !! In symbols !! Derivative in terms of the old functions and their derivatives !! Proof
|-
| [[pointwise sum of functions|pointwise sum]] || <math>f + g</math> is the function <math>x \mapsto f(x) + g(x)</math> <math>f_1 + f_2 + \dots + f_n</math> is the function <math>x \mapsto f_1(x) + f_2(x) + \dots + f_n(x)</math> || Sum of the derivatives of the functions being added (''the derivative of the sum is the sum of the derivatives'') <math>\! f' + g'</math> <math>\! f_1' + f_2' + \dots + f_n'</math> || [[differentiation is linear]]
|-
| [[pointwise difference of functions|pointwise difference]] || <math>f - g</math> is the function <math>x \mapsto f(x) - g(x)</math>|| Difference of the derivatives, i.e., <math>f' - g'</math> || [[differentiation is linear]]
|-
| [[scalar multiple of function|scalar multiple]] by a constant || <math>af</math> is the function <math>x \mapsto af(x)</math> where <math>a</math> is a real number || <math>x \mapsto af'(x)</math> || [[differentiation is linear]]
|-
| [[pointwise product of functions|pointwise product]] || <math>f \cdot g</math> (sometimes denoted <math>fg</math>) is the function <math>x \mapsto f(x)g(x)</math> <math>f_1 \cdot f_2 \cdot \dots f_n</math> (sometimes denoted <math>f_1f_2\dots f_n</math> is the function <math>x \mapsto f_1(x)f_2(x) \dots f_n(x)</math> || For two functions, <math>x \mapsto f'(x)g(x) + f(x)g'(x)</math> For multiple functions, <math>x \mapsto f_1'(x)f_2(x) \dots f_n(x) + f_1(x)f_2'(x) \dots f_n(x) + \dots + f_1(x)f_2(x) \dots f_n'(x)</math> || [[product rule for differentiation]]
|-
| [[pointwise quotient of functions|pointwise quotient]] || <math>f/g</math> is the function <math>x \mapsto f(x)/g(x)</math> || <math>x \mapsto \frac{g(x)f'(x) - f(x)g'(x)}{(g(x))^2}</math> || [[quotient rule for differentiation]]
|-
| [[composite of two functions]] || <math>f \circ g</math> is the function <math>x \mapsto f(g(x))</math> || <math>x \mapsto f'(g(x))g'(x)</math> || [[chain rule for differentiation]]
|-
| [[inverse function]] of a [[one-one function]] || <math>f^{-1}</math> sends <matH>x</math> to the unique <math>y</math> such that <math>f(y) = x</math> || <math>\! \frac{1}{f'(f^{-1}(x))}</math> || [[inverse function theorem]]
|-
| [[piecewise definition of functions|piecewise definition]] || <math>f(x) := \left\lbrace \begin{array}{rl} f_1(x), & x < c \\ f_2(x), & c < x \\v, & x = c \end{array}\right.</math> where <math>f_1, f_2</math> can be extended to differentiable functions on all reals|| <math>f' = f_1'</math> to the left of <math>c</math> and <math>f' = f_2'</math> to the right of <math>c</math>. At <math>c</math>, <math>f</math> is differentiable iff <math>f_1(c) = f_2(c) = v</math> and <math>f_1'(c) = f_2'(c)</math>. || [[differentiation rule for piecewise definition by interval]]
|}

<section end="differentiation rules"/>

=== Significance of differentiation rules ===

The list of differentiation rules is fairly complete with respect to the most typical ways of constructing functions from other functions. This means that if we have explicit expressions for the derivatives of a collection of functions, we can obtain explicit expressions for the derivatives of any other function constructed from them via any of the methods covered in the table above (pointwise sum, pointwise difference, scalar multiple, pointwise product, pointwise quotient, composite, inverse, and piecewise definition).

Computer programs that implement symbolic mathematics (such as Mathematica) generally have all these rules coded in, and are able to use them to differentiate any function constructed by applying these operations to functions whose derivative is already known by the program.

Derivative

2024-04-11T02:39:19Z

Vipul: /* Algebraic definition elaborated in terms of epsilon-delta definition of limits */

{{core term}}

==Name==

The term '''derivative''' is used for the notion defined here. However, there are many variations of the concept of derivative that are described by using adjectives to modify the noun. When these variations are being talked about, it is helpful to provide a similar adjective to indicate that we are talking about the usual notion of derivative. The variations and corresponding terminological clarification are below:

{| class="sortable" border="1"
! Variation of notion of derivative !! Modified name for the usual notion of derivative to emphasize it's the original notion and not the variation
|-
| one-sided derivative (left hand derivative and right hand derivative) -- defined on this page || two-sided derivative
|-
| [[higher derivative]] (obtained by repeated differentiation) || first derivative
|-
| [[partial derivative]] (derivative of a function of multiple variables with respect to one of the variables holding the others constant) || ordinary derivative
|-
| [[discrete derivative]] (not commonly used) || continuous derivative (not commonly used)
|}

==Definition at a point==

===Conceptual definition===

Suppose <math>f</math> is a [[function]] defined on a subset of the reals and <math>x_0</math> is a point in the [[interior]] of the [[domain]] of <math>f</math>, i.e., the domain of <math>f</math> contains an [[open interval]] surrounding <math>x_0</math>. The '''derivative''' of <math>f</math> at <math>x_0</math>, denoted <math>\! f'(x_0)</math>, is the '''instantaneous rate of change''' of <math>f(x)</math> with respect to <math>x</math> at <math>x_0</math>. It is defined as the [[limit]] of the '''average rate of change''' of <math>f</math> between <math>x</math> and <math>x_0</math>, as <math>x</math> approaches <math>x_0</math>.

In the more formal definitions below, we will see that:

* [[Difference quotient]] formalizes the notion of average rate of change.
* Derivative formalizes the notion of instantaneous rate of change, and is the limit of the difference quotient.

===Algebraic definition===

Suppose <math>f</math> is a [[function]] defined on a subset of the reals and <math>x_0</math> is a point in the [[interior]] of the [[domain]] of <math>f</math>, i.e., the domain of <math>f</math> contains an [[open interval]] surrounding <math>x_0</math>. The '''derivative''' (also called '''first derivative''') of <math>f</math> at <math>x_0</math>, denoted <math>f'(x_0)</math> is defined as the [[limit]] of the [[defining ingredient::difference quotient]] of <math>f</math> between <math>x_0</math> and <math>x</math>, as <math>x \to x_0</math>. Explicitly:

<math>\! f'(x_0) := \lim_{x \to x_0} \Delta f(x,x_0) = \lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0}</math>

If this limit exists, then we say that the derivative exists and has this value, and we say that the function is '''differentiable''' at the point. If the limit does not exist, then we say that the function is ''not'' differentiable at the point and the derivative does ''not'' exist.

===Computationally useful version of algebraic definition===

This is obtained from the previous definition by the variable substitution <math>h := x - x_0</math> so <math>x = x_0 + h</math>. Explicitly:

<math>\! f'(x_0) := \lim_{h \to 0} \Delta f(x_0 + h,x_0) = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}</math>

===Geometric definition===

Suppose <math>f</math> is a [[function]] and <math>x_0</math> is a point in the [[interior]] of the [[domain]] of <math>f</math>, i.e., the domain of <math>f</math> contains an [[open interval]] surrounding <math>x_0</math>. The '''derivative''' of <math>f</math> at <math>x_0</math> is the [[slope]] of the [[tangent line]] to the [[graph]] of <math>f</math> through the point <math>(x_0,f(x_0))</math>.

===Algebraic definition elaborated in terms of epsilon-delta definition of limits===

Suppose <math>f</math> is a [[function]] defined on a subset of the reals and <math>x_0</math> is a point in the [[interior]] of the [[domain]] of <math>f</math>, i.e., the domain of <math>f</math> contains an [[open interval]] surrounding <math>x_0</math>. The '''derivative''' (also called '''first derivative''') of <math>f</math> at <math>x_0</math>, denoted <math>\! f'(x_0)</math>, is defined as a real number <math>L</math> such that:

{{quotation|For every <math>\varepsilon > 0</math> there exists <math>\delta > 0</math> such that if <math>\! 0 < |x - x_0| < \delta</math> then <math>|f(x) - f(x_0) - L(x - x_0)| < \varepsilon|x - x_0|</math>.}}

<center>{{#widget:YouTube|id=cZVLsvvBDgo}}</center>

==Definition as a function==

Suppose <math>f</math> is a [[function]] defined on a subset of the reals. Its '''derivative''' or '''first derivative''', denoted <math>f'</math>, is a function defined as follows:

* The domain is the following subset of the domain of <math>f</math>: An element in the domain of <math>f</math> is in the domain of <math>f'</math> if and only if it is in the ''interior'' of the domain of <math>f</math> and the derivative of <math>f</math> exists at the point.
* The function value at any point in the domain is simply the value of the derivative of <math>f</math> at that point.

{{generic point specific point confusion}}

==One-sided notions==

===Left-hand derivative===

Suppose <math>f</math> is a function defined at a point <math>x_0 \in \R</math> and also to the immediate left of <math>x_0</math>. The '''left-hand derivative''' of <math>f</math> at <math>x_0</math> is defined as the ''left hand limit'' for the difference quotient between <math>x</math> and <math>x_0</math>. In other words, it is:

<math>\operatorname{LHD}(f)(x_0) = f'_-(x_0) := \lim_{x \to x_0^-}\Delta f(x,x_0) = \lim_{x \to x_0^-} \frac{f(x) - f(x_0)}{x - x_0} = \lim_{h \to 0^-} \frac{f(x_0 + h) - f(x_0))}{h}</math>

===Right-hand derivative===

Suppose <math>f</math> is a function defined at a point <math>x_0 \in \R</math> and also to the immediate right of <math>x_0</math>. The '''right-hand derivative''' of <math>f</math> at <math>x_0</math> is defined as the ''right hand limit'' for the difference quotient between <math>x</math> and <math>x_0</math>. In other words, it is:

<math>\operatorname{RHD}(f)(x_0) = f'_+(x_0) := \lim_{x \to x_0^+} \Delta f(x,x_0) = \lim_{x \to x_0^+} \frac{f(x) - f(x_0)}{x - x_0} = \lim_{h \to 0^+} \frac{f(x_0 + h) - f(x_0)}{h}</math>

===Relation between one-sided derivatives and the usual (two-sided) derivative===

The derivative <math>f'(x_0)</math> exists if and only if (both the left hand derivative and the right hand derivative exist at <math>x_0</math> and their values are equal). Further, the value of the derivative equals both these equals values.

==Leibniz notation for derivative==

The Leibniz notation for derivative views the derivative as the ''relative rate of change of two variables'' and is thus a somewhat different perspective on the derivative.

Suppose <math>f</math> is a function, and <math>x,y</math> are variables related by <math>y := f(x)</math>. Here, <math>x</math> is an ''independent variable'' and <math>y</math> is the ''dependent variable'' (with the dependency being described by the function <math>f</math>). We then define:

<math>\frac{dy}{dx} := f'(x)</math>

In particular, <math>dy/dx</math> is a ''function'' of <math>x</math>. Its value at <math>x = x_0</math> is defined as <math>f'(x_0)</math> and is denoted as follows:

<math>\! \frac{dy}{dx} |_{x = x_0} := f'(x_0)</math>

Note that the <math>dy/dx</math> notation does ''not'' mean that a number <math>dy</math> is being divided by a number <math>dx</math>. One way of justifying this notation is by expressing it as a limit of a [[difference quotient]]; here, <math>y_0 = f(x_0)</math>:

<math>\! \frac{dy}{dx} |_{x = x_0} := \lim_{x \to x_0} \frac{y - y_0}{x - x_0} = \lim_{x \to x_0} \frac{\Delta y}{\Delta x}</math>

where <math>\Delta y = y - y_0</math> denotes the ''difference'' in <math>y</math>-values and <math>\Delta x = x - x_0</math> denotes the ''difference'' in <math>x</math>-values.

===Quotient notation is misleading but salvageable===

The difference quotient is actually a quotient of ''numbers'', and the derivative is a limit of this. Hence, many of the formal manipulations involving fractions of numbers work with this notation, even though <math>dy/dx</math> ''itself'' is not a quotient of numbers (see [[chain rule for differentiation]] and [[inverse function theorem]]).

===Expressive advantage of Leibniz notation===

The Leibniz notation is advantageous for carrying out computations by hand and writing derivative expressions because it does not require us to ''name'' every function in order to differentiate it. On the other hand, the prime notation requires us to name a function before we can talk of its derivatives.

Thus, the Leibniz notation is crucial for constructing complicated expressions involving derivatives. For instance, consider the expression:

<math>\frac{d}{dx}\left[\left(\frac{d}{dx}(x - \cos x)\right)\sin^2\left(\frac{d}{dx}(x^2 \cos(x^3))\right)\right]</math>

In order to write this expression with the prime notation, we would first need to give names to functions <math>x - \cos x</math> and <math>x^2 \cos x^3</math>, then give a name to the entire expression within square braces, and then talk of differentiating it.

===Expressive disadvantage of Leibniz notation===

The Leibniz notation is not ''point-free'', i.e., we have to use a symbol to denote the point at which the function is being applied. In contrast, with the prime notation, we can make statements like <math>\! \sin' = \cos</math>.

==Physical applications==

===Note on units===

In applications to the natural and social sciences, the ''units'' used for measuring <math>dy/dx</math> are the units used for measuring <math>y</math> divided by the units used for measuring <math>x</math>. This is because the derivative is a limit of a difference quotient, which is a quantity measured in units of <math>y</math> divided by a quantity measured in units of <math>x</math>.

If the dimensions are expressed using a framework such as the MLT framework for physical quantities, then the MLT exponents subtract.

{| class="sortable" border="1"
! Context !! Example of derivative from real world application !! Functionally dependent variable being differentiated !! Independent variable in terms of which differentiation is happening !! Corresponding difference quotient !! Term for numerator of difference quotient !! Term for denominator of difference quotient !! Comment
|-
| kinematics (classical mechanics, physics) || instantaneous velocity measured in units of length/time || position measured in units of length || time measured in units of time || average velocity || displacement || time elapsed || strictly speaking, this is a vector-valued derivative, but we can use single variable calculus if we restrict to motion along a straight line.
|-
| kinematics (classical mechanics, physics) || instantaneous acceleration measured in units of length/(time)^2 || velocity measured in units of length/time || time measured in units of time || average acceleration || change in velocity || time elapsed || strictly speaking, this is a vector-valued derivative, but we can use single variable calculus if we restrict to motion along a straight line.
|-
| chemical reaction (chemistry) || rate of change of concentration of a particular reaction product measured in units of (concentration measure)/(time). || concentration Suitable concentration measure could be molarity (for reactions in solution) or partial pressure (for gaseous reactions) || time measured in units of time || average rate of change of concentration of the product measured in units of (concentration measure)/(time). || change in concentration of product || time elapsed ||
|}

==Related notions==

{| class="sortable" border="1"
! Notion !! How it relates to derivative
|-
| [[higher derivative]] || differentiate again the function obtained by differentiating a particular function, and apply this process repeatedly. Specifically, the <math>k^{th}</math> derivative is the function obtained by applying the differentiation operation <math>k</math> times.
|-
| [[antiderivative]] || a function that has the given derivative. Antidifferentiation is the reverse of differentiation. The general expression for the antiderivative is also called the ''indefinite integral''.
|-
| [[partial derivative]] || a function of more than one variable is differentiated with respect to one of the variables keeping the others constant.
|-
| [[higher partial derivative]] || obtained by applying the partial differentiation operation to a function of more than one variable. The ''pure'' higher partials are those where all the partial differentiation operations are with respect to the same variable. The ''mixed'' higher partials are those where the partial differentiation operations are with respect to more than one variable.
|-
| [[differential]] || {{fillin}}
|}

==Significance==

===Significance of sign on intervals===

The derivative represents the ''rate of change'', and roughly speaking, the sign of derivative represents the ''direction of change''. We list the loose and precise statements below:

{| class="sortable" border="1"
! Loose statement !! Precise versions
|-
| [[increasing function]] means positive derivative || [[positive derivative implies increasing]] [[nonnegative derivative that is not identically zero on any interval implies increasing]] [[increasing and differentiable implies nonnegative derivative that is not identically zero on any interval]]
|-
| [[decreasing function]] means negative derivative || [[negative derivative implies decreasing]] [[decreasing and differentiable implies nonpositive derivative that is not identically zero on any interval]]
|-
| [[constant function]] means zero derivative || [[constant function implies zero derivative]] [[zero derivative implies locally constant]]
|}

===Significance of sign at points===

This is quite similar to the significance on an interval, but the behavior at individual points can be anomalous and can also represent transitions between different kinds of intervals.

{| class="sortable" border="1"
! Loose statement !! Precise versions
|-
| comparison of function value with points on immediate left/right tells us sign of one-sided derivative and vice versa || [[local maximum from the left implies left hand derivative is nonnegative if it exists]] [[local maximum from the right implies right hand derivative is nonpositive if it exists]] [[local minimum from the left implies left hand derivative is nonpositive if it exists]] [[.ocal minimum from the right implies right hand derivative is nonnegative if it exists]]
|-
| local maximum/minimum value must occur at [[critical point]], which is a point where derivative is zero or does not exist. Moreover, sign of derivative on immediate left and right help determine whether it is local max, min, or neither. || [[first derivative test]], see also [[second derivative test]] and [[higher derivative tests]].
|}

==Computation of derivative==

=== List of most commonly used rules ===

<section begin="differentiation rules"/>

For a full list, see [[:Category:Differentiation rules]].

{| class="sortable" border="1"
! Method for constructing new functions from old !! In symbols !! Derivative in terms of the old functions and their derivatives !! Proof
|-
| [[pointwise sum of functions|pointwise sum]] || <math>f + g</math> is the function <math>x \mapsto f(x) + g(x)</math> <math>f_1 + f_2 + \dots + f_n</math> is the function <math>x \mapsto f_1(x) + f_2(x) + \dots + f_n(x)</math> || Sum of the derivatives of the functions being added (''the derivative of the sum is the sum of the derivatives'') <math>\! f' + g'</math> <math>\! f_1' + f_2' + \dots + f_n'</math> || [[differentiation is linear]]
|-
| [[pointwise difference of functions|pointwise difference]] || <math>f - g</math> is the function <math>x \mapsto f(x) - g(x)</math>|| Difference of the derivatives, i.e., <math>f' - g'</math> || [[differentiation is linear]]
|-
| [[scalar multiple of function|scalar multiple]] by a constant || <math>af</math> is the function <math>x \mapsto af(x)</math> where <math>a</math> is a real number || <math>x \mapsto af'(x)</math> || [[differentiation is linear]]
|-
| [[pointwise product of functions|pointwise product]] || <math>f \cdot g</math> (sometimes denoted <math>fg</math>) is the function <math>x \mapsto f(x)g(x)</math> <math>f_1 \cdot f_2 \cdot \dots f_n</math> (sometimes denoted <math>f_1f_2\dots f_n</math> is the function <math>x \mapsto f_1(x)f_2(x) \dots f_n(x)</math> || For two functions, <math>x \mapsto f'(x)g(x) + f(x)g'(x)</math> For multiple functions, <math>x \mapsto f_1'(x)f_2(x) \dots f_n(x) + f_1(x)f_2'(x) \dots f_n(x) + \dots + f_1(x)f_2(x) \dots f_n'(x)</math> || [[product rule for differentiation]]
|-
| [[pointwise quotient of functions|pointwise quotient]] || <math>f/g</math> is the function <math>x \mapsto f(x)/g(x)</math> || <math>x \mapsto \frac{g(x)f'(x) - f(x)g'(x)}{(g(x))^2}</math> || [[quotient rule for differentiation]]
|-
| [[composite of two functions]] || <math>f \circ g</math> is the function <math>x \mapsto f(g(x))</math> || <math>x \mapsto f'(g(x))g'(x)</math> || [[chain rule for differentiation]]
|-
| [[inverse function]] of a [[one-one function]] || <math>f^{-1}</math> sends <matH>x</math> to the unique <math>y</math> such that <math>f(y) = x</math> || <math>\! \frac{1}{f'(f^{-1}(x))}</math> || [[inverse function theorem]]
|-
| [[piecewise definition of functions|piecewise definition]] || <math>f(x) := \left\lbrace \begin{array}{rl} f_1(x), & x < c \\ f_2(x), & c < x \\v, & x = c \end{array}\right.</math> where <math>f_1, f_2</math> can be extended to differentiable functions on all reals|| <math>f' = f_1'</math> to the left of <math>c</math> and <math>f' = f_2'</math> to the right of <math>c</math>. At <math>c</math>, <math>f</math> is differentiable iff <math>f_1(c) = f_2(c) = v</math> and <math>f_1'(c) = f_2'(c)</math>. || [[differentiation rule for piecewise definition by interval]]
|}

<section end="differentiation rules"/>

=== Significance of differentiation rules ===

The list of differentiation rules is fairly complete with respect to the most typical ways of constructing functions from other functions. This means that if we have explicit expressions for the derivatives of a collection of functions, we can obtain explicit expressions for the derivatives of any other function constructed from them via any of the methods covered in the table above (pointwise sum, pointwise difference, scalar multiple, pointwise product, pointwise quotient, composite, inverse, and piecewise definition).

Computer programs that implement symbolic mathematics (such as Mathematica) generally have all these rules coded in, and are able to use them to differentiate any function constructed by applying these operations to functions whose derivative is already known by the program.

Derivative

2024-04-11T02:36:15Z

Vipul: /* Definition as a function */

{{core term}}

==Name==

The term '''derivative''' is used for the notion defined here. However, there are many variations of the concept of derivative that are described by using adjectives to modify the noun. When these variations are being talked about, it is helpful to provide a similar adjective to indicate that we are talking about the usual notion of derivative. The variations and corresponding terminological clarification are below:

{| class="sortable" border="1"
! Variation of notion of derivative !! Modified name for the usual notion of derivative to emphasize it's the original notion and not the variation
|-
| one-sided derivative (left hand derivative and right hand derivative) -- defined on this page || two-sided derivative
|-
| [[higher derivative]] (obtained by repeated differentiation) || first derivative
|-
| [[partial derivative]] (derivative of a function of multiple variables with respect to one of the variables holding the others constant) || ordinary derivative
|-
| [[discrete derivative]] (not commonly used) || continuous derivative (not commonly used)
|}

==Definition at a point==

===Conceptual definition===

Suppose <math>f</math> is a [[function]] defined on a subset of the reals and <math>x_0</math> is a point in the [[interior]] of the [[domain]] of <math>f</math>, i.e., the domain of <math>f</math> contains an [[open interval]] surrounding <math>x_0</math>. The '''derivative''' of <math>f</math> at <math>x_0</math>, denoted <math>\! f'(x_0)</math>, is the '''instantaneous rate of change''' of <math>f(x)</math> with respect to <math>x</math> at <math>x_0</math>. It is defined as the [[limit]] of the '''average rate of change''' of <math>f</math> between <math>x</math> and <math>x_0</math>, as <math>x</math> approaches <math>x_0</math>.

In the more formal definitions below, we will see that:

* [[Difference quotient]] formalizes the notion of average rate of change.
* Derivative formalizes the notion of instantaneous rate of change, and is the limit of the difference quotient.

===Algebraic definition===

Suppose <math>f</math> is a [[function]] defined on a subset of the reals and <math>x_0</math> is a point in the [[interior]] of the [[domain]] of <math>f</math>, i.e., the domain of <math>f</math> contains an [[open interval]] surrounding <math>x_0</math>. The '''derivative''' (also called '''first derivative''') of <math>f</math> at <math>x_0</math>, denoted <math>f'(x_0)</math> is defined as the [[limit]] of the [[defining ingredient::difference quotient]] of <math>f</math> between <math>x_0</math> and <math>x</math>, as <math>x \to x_0</math>. Explicitly:

<math>\! f'(x_0) := \lim_{x \to x_0} \Delta f(x,x_0) = \lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0}</math>

If this limit exists, then we say that the derivative exists and has this value, and we say that the function is '''differentiable''' at the point. If the limit does not exist, then we say that the function is ''not'' differentiable at the point and the derivative does ''not'' exist.

===Computationally useful version of algebraic definition===

This is obtained from the previous definition by the variable substitution <math>h := x - x_0</math> so <math>x = x_0 + h</math>. Explicitly:

<math>\! f'(x_0) := \lim_{h \to 0} \Delta f(x_0 + h,x_0) = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}</math>

===Geometric definition===

Suppose <math>f</math> is a [[function]] and <math>x_0</math> is a point in the [[interior]] of the [[domain]] of <math>f</math>, i.e., the domain of <math>f</math> contains an [[open interval]] surrounding <math>x_0</math>. The '''derivative''' of <math>f</math> at <math>x_0</math> is the [[slope]] of the [[tangent line]] to the [[graph]] of <math>f</math> through the point <math>(x_0,f(x_0))</math>.

===Algebraic definition elaborated in terms of epsilon-delta definition of limits===

Suppose <math>f</math> is a [[function]] defined on a subset of the reals and <math>x_0</math> is a point in the [[interior]] of the [[domain]] of <math>f</math>, i.e., the domain of <math>f</math> contains an [[open interval]] surrounding <math>x_0</math>. The '''derivative''' (also called '''first derivative''') of <math>f</math> at <math>x_0</math>, denoted <math>\! f'(x_0)</math>, is defined as a real number <math>L</math> such that:

{{quotation|For every <math>\!\varepsilon > 0</math> there exists <math>\!\delta > 0</math> such that if <math>\! 0 < |x - x_0| < \delta</math> then <math>\! |f(x) - f(x_0) - L(x - x_0)| < \varepsilon|x - x_0|</math>.}}

<center>{{#widget:YouTube|id=cZVLsvvBDgo}}</center>

==Definition as a function==

Suppose <math>f</math> is a [[function]] defined on a subset of the reals. Its '''derivative''' or '''first derivative''', denoted <math>f'</math>, is a function defined as follows:

* The domain is the following subset of the domain of <math>f</math>: An element in the domain of <math>f</math> is in the domain of <math>f'</math> if and only if it is in the ''interior'' of the domain of <math>f</math> and the derivative of <math>f</math> exists at the point.
* The function value at any point in the domain is simply the value of the derivative of <math>f</math> at that point.

{{generic point specific point confusion}}

==One-sided notions==

===Left-hand derivative===

Suppose <math>f</math> is a function defined at a point <math>x_0 \in \R</math> and also to the immediate left of <math>x_0</math>. The '''left-hand derivative''' of <math>f</math> at <math>x_0</math> is defined as the ''left hand limit'' for the difference quotient between <math>x</math> and <math>x_0</math>. In other words, it is:

<math>\operatorname{LHD}(f)(x_0) = f'_-(x_0) := \lim_{x \to x_0^-}\Delta f(x,x_0) = \lim_{x \to x_0^-} \frac{f(x) - f(x_0)}{x - x_0} = \lim_{h \to 0^-} \frac{f(x_0 + h) - f(x_0))}{h}</math>

===Right-hand derivative===

Suppose <math>f</math> is a function defined at a point <math>x_0 \in \R</math> and also to the immediate right of <math>x_0</math>. The '''right-hand derivative''' of <math>f</math> at <math>x_0</math> is defined as the ''right hand limit'' for the difference quotient between <math>x</math> and <math>x_0</math>. In other words, it is:

<math>\operatorname{RHD}(f)(x_0) = f'_+(x_0) := \lim_{x \to x_0^+} \Delta f(x,x_0) = \lim_{x \to x_0^+} \frac{f(x) - f(x_0)}{x - x_0} = \lim_{h \to 0^+} \frac{f(x_0 + h) - f(x_0)}{h}</math>

===Relation between one-sided derivatives and the usual (two-sided) derivative===

The derivative <math>f'(x_0)</math> exists if and only if (both the left hand derivative and the right hand derivative exist at <math>x_0</math> and their values are equal). Further, the value of the derivative equals both these equals values.

==Leibniz notation for derivative==

The Leibniz notation for derivative views the derivative as the ''relative rate of change of two variables'' and is thus a somewhat different perspective on the derivative.

Suppose <math>f</math> is a function, and <math>x,y</math> are variables related by <math>y := f(x)</math>. Here, <math>x</math> is an ''independent variable'' and <math>y</math> is the ''dependent variable'' (with the dependency being described by the function <math>f</math>). We then define:

<math>\frac{dy}{dx} := f'(x)</math>

In particular, <math>dy/dx</math> is a ''function'' of <math>x</math>. Its value at <math>x = x_0</math> is defined as <math>f'(x_0)</math> and is denoted as follows:

<math>\! \frac{dy}{dx} |_{x = x_0} := f'(x_0)</math>

Note that the <math>dy/dx</math> notation does ''not'' mean that a number <math>dy</math> is being divided by a number <math>dx</math>. One way of justifying this notation is by expressing it as a limit of a [[difference quotient]]; here, <math>y_0 = f(x_0)</math>:

<math>\! \frac{dy}{dx} |_{x = x_0} := \lim_{x \to x_0} \frac{y - y_0}{x - x_0} = \lim_{x \to x_0} \frac{\Delta y}{\Delta x}</math>

where <math>\Delta y = y - y_0</math> denotes the ''difference'' in <math>y</math>-values and <math>\Delta x = x - x_0</math> denotes the ''difference'' in <math>x</math>-values.

===Quotient notation is misleading but salvageable===

The difference quotient is actually a quotient of ''numbers'', and the derivative is a limit of this. Hence, many of the formal manipulations involving fractions of numbers work with this notation, even though <math>dy/dx</math> ''itself'' is not a quotient of numbers (see [[chain rule for differentiation]] and [[inverse function theorem]]).

===Expressive advantage of Leibniz notation===

The Leibniz notation is advantageous for carrying out computations by hand and writing derivative expressions because it does not require us to ''name'' every function in order to differentiate it. On the other hand, the prime notation requires us to name a function before we can talk of its derivatives.

Thus, the Leibniz notation is crucial for constructing complicated expressions involving derivatives. For instance, consider the expression:

<math>\frac{d}{dx}\left[\left(\frac{d}{dx}(x - \cos x)\right)\sin^2\left(\frac{d}{dx}(x^2 \cos(x^3))\right)\right]</math>

In order to write this expression with the prime notation, we would first need to give names to functions <math>x - \cos x</math> and <math>x^2 \cos x^3</math>, then give a name to the entire expression within square braces, and then talk of differentiating it.

===Expressive disadvantage of Leibniz notation===

The Leibniz notation is not ''point-free'', i.e., we have to use a symbol to denote the point at which the function is being applied. In contrast, with the prime notation, we can make statements like <math>\! \sin' = \cos</math>.

==Physical applications==

===Note on units===

In applications to the natural and social sciences, the ''units'' used for measuring <math>dy/dx</math> are the units used for measuring <math>y</math> divided by the units used for measuring <math>x</math>. This is because the derivative is a limit of a difference quotient, which is a quantity measured in units of <math>y</math> divided by a quantity measured in units of <math>x</math>.

If the dimensions are expressed using a framework such as the MLT framework for physical quantities, then the MLT exponents subtract.

{| class="sortable" border="1"
! Context !! Example of derivative from real world application !! Functionally dependent variable being differentiated !! Independent variable in terms of which differentiation is happening !! Corresponding difference quotient !! Term for numerator of difference quotient !! Term for denominator of difference quotient !! Comment
|-
| kinematics (classical mechanics, physics) || instantaneous velocity measured in units of length/time || position measured in units of length || time measured in units of time || average velocity || displacement || time elapsed || strictly speaking, this is a vector-valued derivative, but we can use single variable calculus if we restrict to motion along a straight line.
|-
| kinematics (classical mechanics, physics) || instantaneous acceleration measured in units of length/(time)^2 || velocity measured in units of length/time || time measured in units of time || average acceleration || change in velocity || time elapsed || strictly speaking, this is a vector-valued derivative, but we can use single variable calculus if we restrict to motion along a straight line.
|-
| chemical reaction (chemistry) || rate of change of concentration of a particular reaction product measured in units of (concentration measure)/(time). || concentration Suitable concentration measure could be molarity (for reactions in solution) or partial pressure (for gaseous reactions) || time measured in units of time || average rate of change of concentration of the product measured in units of (concentration measure)/(time). || change in concentration of product || time elapsed ||
|}

==Related notions==

{| class="sortable" border="1"
! Notion !! How it relates to derivative
|-
| [[higher derivative]] || differentiate again the function obtained by differentiating a particular function, and apply this process repeatedly. Specifically, the <math>k^{th}</math> derivative is the function obtained by applying the differentiation operation <math>k</math> times.
|-
| [[antiderivative]] || a function that has the given derivative. Antidifferentiation is the reverse of differentiation. The general expression for the antiderivative is also called the ''indefinite integral''.
|-
| [[partial derivative]] || a function of more than one variable is differentiated with respect to one of the variables keeping the others constant.
|-
| [[higher partial derivative]] || obtained by applying the partial differentiation operation to a function of more than one variable. The ''pure'' higher partials are those where all the partial differentiation operations are with respect to the same variable. The ''mixed'' higher partials are those where the partial differentiation operations are with respect to more than one variable.
|-
| [[differential]] || {{fillin}}
|}

==Significance==

===Significance of sign on intervals===

The derivative represents the ''rate of change'', and roughly speaking, the sign of derivative represents the ''direction of change''. We list the loose and precise statements below:

{| class="sortable" border="1"
! Loose statement !! Precise versions
|-
| [[increasing function]] means positive derivative || [[positive derivative implies increasing]] [[nonnegative derivative that is not identically zero on any interval implies increasing]] [[increasing and differentiable implies nonnegative derivative that is not identically zero on any interval]]
|-
| [[decreasing function]] means negative derivative || [[negative derivative implies decreasing]] [[decreasing and differentiable implies nonpositive derivative that is not identically zero on any interval]]
|-
| [[constant function]] means zero derivative || [[constant function implies zero derivative]] [[zero derivative implies locally constant]]
|}

===Significance of sign at points===

This is quite similar to the significance on an interval, but the behavior at individual points can be anomalous and can also represent transitions between different kinds of intervals.

{| class="sortable" border="1"
! Loose statement !! Precise versions
|-
| comparison of function value with points on immediate left/right tells us sign of one-sided derivative and vice versa || [[local maximum from the left implies left hand derivative is nonnegative if it exists]] [[local maximum from the right implies right hand derivative is nonpositive if it exists]] [[local minimum from the left implies left hand derivative is nonpositive if it exists]] [[.ocal minimum from the right implies right hand derivative is nonnegative if it exists]]
|-
| local maximum/minimum value must occur at [[critical point]], which is a point where derivative is zero or does not exist. Moreover, sign of derivative on immediate left and right help determine whether it is local max, min, or neither. || [[first derivative test]], see also [[second derivative test]] and [[higher derivative tests]].
|}

==Computation of derivative==

=== List of most commonly used rules ===

<section begin="differentiation rules"/>

For a full list, see [[:Category:Differentiation rules]].

{| class="sortable" border="1"
! Method for constructing new functions from old !! In symbols !! Derivative in terms of the old functions and their derivatives !! Proof
|-
| [[pointwise sum of functions|pointwise sum]] || <math>f + g</math> is the function <math>x \mapsto f(x) + g(x)</math> <math>f_1 + f_2 + \dots + f_n</math> is the function <math>x \mapsto f_1(x) + f_2(x) + \dots + f_n(x)</math> || Sum of the derivatives of the functions being added (''the derivative of the sum is the sum of the derivatives'') <math>\! f' + g'</math> <math>\! f_1' + f_2' + \dots + f_n'</math> || [[differentiation is linear]]
|-
| [[pointwise difference of functions|pointwise difference]] || <math>f - g</math> is the function <math>x \mapsto f(x) - g(x)</math>|| Difference of the derivatives, i.e., <math>f' - g'</math> || [[differentiation is linear]]
|-
| [[scalar multiple of function|scalar multiple]] by a constant || <math>af</math> is the function <math>x \mapsto af(x)</math> where <math>a</math> is a real number || <math>x \mapsto af'(x)</math> || [[differentiation is linear]]
|-
| [[pointwise product of functions|pointwise product]] || <math>f \cdot g</math> (sometimes denoted <math>fg</math>) is the function <math>x \mapsto f(x)g(x)</math> <math>f_1 \cdot f_2 \cdot \dots f_n</math> (sometimes denoted <math>f_1f_2\dots f_n</math> is the function <math>x \mapsto f_1(x)f_2(x) \dots f_n(x)</math> || For two functions, <math>x \mapsto f'(x)g(x) + f(x)g'(x)</math> For multiple functions, <math>x \mapsto f_1'(x)f_2(x) \dots f_n(x) + f_1(x)f_2'(x) \dots f_n(x) + \dots + f_1(x)f_2(x) \dots f_n'(x)</math> || [[product rule for differentiation]]
|-
| [[pointwise quotient of functions|pointwise quotient]] || <math>f/g</math> is the function <math>x \mapsto f(x)/g(x)</math> || <math>x \mapsto \frac{g(x)f'(x) - f(x)g'(x)}{(g(x))^2}</math> || [[quotient rule for differentiation]]
|-
| [[composite of two functions]] || <math>f \circ g</math> is the function <math>x \mapsto f(g(x))</math> || <math>x \mapsto f'(g(x))g'(x)</math> || [[chain rule for differentiation]]
|-
| [[inverse function]] of a [[one-one function]] || <math>f^{-1}</math> sends <matH>x</math> to the unique <math>y</math> such that <math>f(y) = x</math> || <math>\! \frac{1}{f'(f^{-1}(x))}</math> || [[inverse function theorem]]
|-
| [[piecewise definition of functions|piecewise definition]] || <math>f(x) := \left\lbrace \begin{array}{rl} f_1(x), & x < c \\ f_2(x), & c < x \\v, & x = c \end{array}\right.</math> where <math>f_1, f_2</math> can be extended to differentiable functions on all reals|| <math>f' = f_1'</math> to the left of <math>c</math> and <math>f' = f_2'</math> to the right of <math>c</math>. At <math>c</math>, <math>f</math> is differentiable iff <math>f_1(c) = f_2(c) = v</math> and <math>f_1'(c) = f_2'(c)</math>. || [[differentiation rule for piecewise definition by interval]]
|}

<section end="differentiation rules"/>

=== Significance of differentiation rules ===

The list of differentiation rules is fairly complete with respect to the most typical ways of constructing functions from other functions. This means that if we have explicit expressions for the derivatives of a collection of functions, we can obtain explicit expressions for the derivatives of any other function constructed from them via any of the methods covered in the table above (pointwise sum, pointwise difference, scalar multiple, pointwise product, pointwise quotient, composite, inverse, and piecewise definition).

Computer programs that implement symbolic mathematics (such as Mathematica) generally have all these rules coded in, and are able to use them to differentiate any function constructed by applying these operations to functions whose derivative is already known by the program.

Differential equation

2024-04-11T02:30:18Z

Vipul: /* Relation with system of first-order differential equations */

==Definition==

===Formal description===

The term '''differential equation''', sometimes called '''ordinary differential equation''' to distinguish it from [[partial differential equation]]s and other variants, is an equation involving two variables, an ''independent variable'' <math>x</math> and a ''dependent'' variable <math>y</math>, as well as the [[derivative]]s (first and possibly higher) of <math>y</math> with respect to <math>x</math>. Formally, it is an equation of the form:

<math>F(x,y,y',y'',\dots,y^{(k)}) = 0</math>

where <math>F</math> is a function of <math>k + 2</math> variables. Here <math>k \ge 1</math>. Note that <math>F</math> may choose not to use some of the derivatives.

In functional notation, the same differential equation may be written as:

<math>F(x,f(x),f'(x),f''(x),\dots,f^{(k)}(x)) = 0</math>

where <math>f</math> is the function such that <math>y = f(x)</math>.

===Solution concept===

* '''Functional solution''': A function <math>f</math> on the domain of interest is said to be a ''solution'' (or ''functional solution'') to the equation if, when we plug in <math>y = f(x)</math>, the equation holds true for ''all'' <math>x</math> in the domain, i.e.:

<math>F(x,f(x),f'(x),f''(x),\dots,f^{(k)}(x)) = 0 \ \forall \ x \in \operatorname{dom}(f)</math>

Note that in cases of functions defined on closed intervals, we exclude checking the conditions on the boundary of the domain because two-sided derivatives don't make sense at the boundary.

* '''Relational solution''': A relation <math>R(x,y) = 0</math> is termed a ''relational solution'' to the equation if <math>F(x,y,y',y'',\dots,y^{(k)}) = 0</math> holds true for all <math>x,y</math> if we calculate the derivatives of <math>y</math> with respect to <math>x</math> using [[implicit differentiation]].

===Initial value problem===

An ''initial value problem'' is a differential equation:

<math>F(x,y,y',y'',\dots,y^{(k)}) = 0</math>

accompanied with a tuple <math>(x_0,y_0,y_1,\dots,y_{k-1})</math>.

A '''functional solution''' to the initial value problem is a functional solution <math>y = f(x)</math> for the differential equation such that <math>f(x_0) = y_0</math> and <math>f^{(i)}(x_0) = y_i</math> for <math>i \in \{ 1,2,\dots,k-1\}</math>.

Analogously, we can define a relational solution to the initial value problem.

==Key observations==

===Differential equations are functional equations===

Differential equations are examples of [[functional equation]]s. A functional equation is an equation where the ''variable'' that we are trying to solve for is a function, and the equation holds true for all values of the input to the function. For instance, here is an example of a functional equation (that's not a differential equation):

<math>f(x + y) = f(x) + f(y) \ \forall \ x,y \in \R</math>

A ''solution'' to a functional equation is a function that satisfies the equation for all choices of inputs. For instance, any function of the form <math>f(x) := ax</math> for fixed <math>a \in \R</math> is a solution to the above functional equation.

Differential equations are functional equations -- we are trying to solve a differential equation, not for the variables, but for the ''functional'' relationship between them.

===Differential equations capture behavior at a single point===

Not every functional equation involving derivatives is a differential equation. Differential equations are characterized by the evaluation of the function and its derivatives all happening at a single point. For instance:

* <math>x^2 = f(x) + (f'(x))^3</math> is a differential equation because all the function and derivative evaluations happen at <math>x</math>, but
* <math>x^2 = f(x) + f'(1 - x)</math> is ''not'' a differential equation in our sense of the word because the derivative evaluation happens at <math>1 - x</math> rather than <math>x</math>.

Another way of putting this is that differential equations are inherently local and cannot relate the behavior of the function at far-away points.

Functional equations involving derivatives that do ''not'' fit this definition of differential equation are also studied, but the study of these is more complicated and requires new techniques. [[Delay differential equation]]s is one such class of functional equations.

===It does not make sense to ask whether a point satisfies a differential equation===

Consider a differential equation:

<math>x^2 = y + y'</math>

If I ask the question: ''does the point <math>x = 2,y = 3</math> satisfy the differential equation?'', the answer is that the question doesn't make any sense. This is because verifying a differential equation requires knowing the ''functional relationship'' between <math>x</math> and <math>y</matH>, which in turn allows us to compute the numerical value of <math>y'</math> and check whether the equation is satisfied.

==Terminology==

===Equation terminology===

{| class="sortable" border="1"
! Term !! Meaning !! Example (don't try to solve these differential equations!)
|-
| [[order of a differential equation]] || it is the largest <math>k</math> for which the <math>k^{th}</math> derivative of the dependent variable appears in the differential equation. || The equation <math>y + xy''' + (y'')^2 = \sin(y')</math> has order three because <math>y'''</math> is the largest derivative appearing.
|-
| [[first-order differential equation]] || differential equation of order one, i.e., it involves only <math>x,y,y'</math>. || <math>y' = \sin(x + yy')</math> is a first-order differential equation.
|-
| [[second-order differential equation]] || differential equation of order exactly two, i.e., it involves only <math>x,y,y',y''</math> and has at least one appearance of <math>y''</math>. || <math>xy'' + \cos(y^2y') = x^2e^{yy'}</math> is a second-order differential equation.
|-
| [[degree of a differential equation]] || if the differential equation is polynomial in terms of its highest order derivative, then the degree of that polynomial. || <math>(y''')^2y' + (y')^5 = 3xy</math> has degree two.
|-
| [[explicit differential equation]] || This means that the highest order derivative is written explicitly in terms of the dependent variable, independent variable, and the lower order derivatives. Any explicit differential equation is a first-degree differential equation. Conversely, any first-degree differential equation can be converted to an explicit differential equation by dividing out by the coefficient of the highest order derivative -- if this coefficient is not invertible, we may separately need to consider the case where that coefficient becomes zero, and that would be a ''lower'' order differential equation. || <math>y''' = xy'' - x^2\sin(yy') + y^3(y'')^5</math> is explicit: the third derivative is written in terms of the lower order derivatives.
|-
| [[autonomous differential equation]] || differential equation where the independent variable does ''not'' appear explicitly anywhere in the equation. || <math>y + y'' = \cos(yy'y''')</math> is autonomous. On the other hand, <math>y + xy' = y''</math> is ''not'' autonomous
|-
| [[linear differential equation]] || A differential equation of the form <math>p_k(x)y^{(k)} + p_{k-1}(x)y^{(k-1)} + \dots + p_0(x)y = q(x)</math> where the <math>p_i</math>s and <math>q</math> are all functions. In other words, the expression is linear in <math>y</math> and its derivatives with coefficients in terms of <math>x</math>. Linear differential equations are usually written with the coefficient of <math>y^{(k)}</math> cleared to 1, by dividing throughout by the coefficient of <math>y^{(k)}</math>. || <math>e^xy''' + x^2y'' + \sin(x)y' + 3y = x^2 - 2x + 5</math> is linear.
|-
| [[homogeneous linear differential equation]] || A linear differential equation of the form <math>p_k(x)y^{(k)} + p_{k-1}(x)y^{(k-1)} + \dots + p_0(x)y = 0</math>. In other words, the ''constant term'' function is zero. || <math>e^xy^{(4)} - 3x^3y''' + \sin(\sin x)y = 0</math> is homogeneous linear.
|-
| [[linear differential equation with constant coefficients]] || A linear differential equation of the form <math>a_ky^{(k)} + a_{k-1}y^{(k-1)} + \dots + a_1y' + a_0y = b</math> where all the <math>a_i</math>s and <math>b</math> are zero. || <math>2y^{(5)} - y^{(3)} + y = 13</math> is linear with constant coefficients.
|}

===Solution terminology===

{| class="sortable" border="1"
! Term !! Meaning !! Example
|-
| particular solution || a function or relation that is a solution for the equation (see [[#Solution concept]]). A solution in the form of a function <math>y = f(x)</math> is termed a ''functional solution'' and a solution in the form of a relation <math>R(x,y) = 0</math> is termed a ''relational solution''. || <math>y = \sin x</math> is a functional solution to <math>y^2 + y'^2 = 1</math>.
|-
| solution family || a family of functions or relations, with one or more parameters possibly subject to some constraints, such that for every choice of parameter values subject to those constraints, we get a particular solution. || <math>y = \sin(x + C)</math> with parameter <math>C \in \R</math>, is a solution family for <math>y^2 + y'^2 = 1</math>.
|-
| general solution || a solution family that covers ''all'' solutions (or almost all solutions, possibly excluding some exceptions) || The general solution to <math>y' = 0</math> is <math>y = C, C \in \R</math>.
|-
| solution to initial value problem || a particular solution that satisfies the initial value condition. || A particular solution to <math>y + y' + y'' = (x + 1)^2</math> satisfying <math>y(0) = -1, y'(0) = 0</math> (i.e., <math>y = -1</math> and <math>y' = 0</math> when <math>x = 0</math>) is <math>y = x^2 - 1</math>.
|}

==Facts==

* As a general principle, the way to solve a differential equation of order <math>k</math> is to reduce it to a sequence of <math>k</math> integration problems. Each integration problem introduces a new freely varying parameter.
* As a general principle, the number of degrees of freedom (i.e., the number of independent freely varying parameters) in the general solution to a differential equation of order <math>k</math> must equal <math>k</math>. There are various exceptions and irregularities, but this is what we should generally expect. Another way of putting this is that the solution space to a differential equation of order <math>k</math> is expected to be <math>k</math>-dimensional.
* As a general principle, the number of solutions to an initial value problem should be finite. If the differential equation is nice enough, then there should be a ''unique'' solution to any initial value problem.

==Relation with system of first-order differential equations==

Any differential equation of order <math>k</math> can be converted to a [[system of first-order differential equations]] with <math>k</math> equations and <math>k</math> variables (i.e., <math>k</math> unknown functions that we are trying to solve for). For the conversion procedure, see [[conversion of a differential equation to a system of first-order differential equations]]. However, the converse is not true, i.e., it is not always possible to convert a system of first-order differential equations with multiple dependent variables into a single differential equation of higher order with one dependent variable.

The same idea can be used to perform the [[conversion of a system of differential equations to a system of first-order differential equations]].

==Solution strategies==

===General idea for strategy toward a general solution===

The rough idea is to convert the differential equation to a sequence of integration problems. If the differential equation has order <math>k</math>, it should reduce to performing <math>k</math> integrations. Each integration introduces a new freely varying parameter.

{{quotation|'''CAUTION''': There may be other auxiliary integrations that need to be done, e.g., for computation of integrating factors or in order to solve an [[integration by parts]] problem. These auxiliary antiderivative computations do not, however, introduce new freely varying parameters.}}

We see that this general strategy may run into trouble at many levels:

{| class="sortable" border="1"
! Stage !! Type of difficulty
|-
| Differential equation to integration || There is no general-purpose algorithm for converting an arbitrary differential equation to an integration problem or sequence of integration problems. Thus, we may not even be able to get started on the process. For some types of structures of differential equations, strategies are known for converting them to integration problems. For others, we have to use ''ad hoc'' techniques. There exist differential equations for which there is no way of converting them to integration problems. Note that for differential equations of order two or higher, this problem may occur at ''any'' of the stages, i.e., we may be able to do one level of integration but not the next.
|-
| Solving the integration problem to get rid of the integral sign || Even after we've converted the differential equation to an integration problem, there may not be any analytic methods for computing the antiderivative. Note that this is not such a big issue because there are known techniques for calculating approximate solutions to integration problems even if analytical methods are not available for getting a precise solution.
|-
| Making sense of relational solutions, converting to functional solutions or at least trying to understand what they mean || Even after we have done the desired antidifferentiations, the solutions may be in relational form rather than functional form. This means that we may still not have ''explicit'' functional descriptions of <math>y</math> in terms of <math>x</math>. We may not even know whether <math>y</math> is expressible as a function of <math>x</math>. We may not even know if the relational solution has any points in it at all!
|}

Historically, this type of method of solution of a differential equation is called a ''solution by quadratures''. In the early days of differential equations, it was hoped that generic differential equations could be solved by quadratures, but this hope was dashed fairly quickly.

===Going from a particular solution to a general solution===

There are some special types of differential equations where, once we find a particular solution, we can find other solutions, perhaps even the general solution. Some notable cases are considered here:

* For a [[linear differential equation]], finding a general solution is equivalent to finding a particular solution + solving the corresponding [[homogeneous linear differential equation]]. In particular, for a [[linear differential equation with constant coefficients]], there is a closed form expression for the solution of the corresponding homogeneous linear differential equation with constant coefficients, so finding a particular solution is equivalent to finding the general solution.
* For an [[autonomous differential equation]], if <math>y = f(x)</math> is a solution, so is <math>y = f(x + C)</math> for any constant <math>C</math>. Note that this does not give the ''general solution'' if the order is more than one, but it does help move from a particular solution to a solution family with one parameter.

===Solution strategies in particular cases===

Below are some formats of equations for which general strategies are known. Note that the letter <math>f</math> is no longer used for the solution function but may be used for other functions.:

{| class="sortable" border="1"
! Equation type !! Order !! Degree (if polynomial in highest order derivative) !! Quick summary of solution strategy
|-
| [[first-order linear differential equation]] which in simplified form looks like <math>y' + p(x)y = q(x)</math> || 1 || 1 || Use the [[integrating factor]] <math>e^{H(x)}</math> where <math>H'=p</math>. The general solution is <math>y = Ce^{-H(x)} + e^{-H(x)}\int p(x)e^{H(x)} \, dx</math>
|-
| [[separable differential equation]] which is of the form <math>y' = f(x)g(y)</math> (any [[first-order first-degree autonomous differential equation]] is separable, though there are separable differential equations that aren't autonomous) || 1 || 1 || Separate and solve as <math>\int \frac{dy}{g(y)} = \int f(x) \, dx</math>. Also find solutions corresponding to <math>y = k</math> where <math>g(k) = 0</math>.
|-
| [[first-order exact differential equation]] <math>F(x,y,y') = 0</math> || 1 || 1 || Try to find a relation <math>R(x,y)</math> such that <math>F(x,y,y') = \frac{d}{dx}[R(x,y)]</math> using [[implicit differentiation]]. Finding the <math>R</math>, even if it does exist, can be tricky.
|-
| [[Bernoulli differential equation]] <math>y' + p(x)y = q(x)y^n</math> (<math>n \ne 0,1</math>) || 1 || 1 || Divide both sides by <math>y^n</math> (set aside possible stationary solution <math>y = 0</math>), then substitute <math>w = 1/y^{n-1}</math> to get a [[first-order linear differential equation]] with dependent variable <math>w</math> and independent variable <math>x</math>.
|-
| [[Clairaut's equation]] which is of the form <math>y = xy' + f(y')</math> || 1 || need not be polynomial; if polynomial, may have any degree || <math>y = Cx + f(C)</math> with <math>C \in \R</math> (all straight lines) and a single other solution explicitly described as the solution to <math>x + f(dy/dx) = 0</math>, given by <math>x = -f'(p), y = f(p) - pf'(p)</math> as a parametric curve in terms of <math>p</math>.
|-
| [[Lagrange equation]] <math>y = f(y')x + g(y')</math> which is linear in <math>x</math> and <math>y</math> but not necessarily in <math>y'</math> || 1 || need not be polynomial; if polynomial, may have any degree || General solution is a family of curves, each described as a parametric curve with parameter the derivative <math>y'</math> (which we denote by <math>p</math>). There may be some special straight line solutions of the form <math>y = px + g(p)</math> for values <math>p</math> with <math>p = f(p)</math>.
|-
| [[second-order autonomous differential equation of degree one]], which is of the form <math>y'' = F(y,y')</math> || 2 || 1 ||
|-
| [[homogeneous linear differential equation with constant coefficients]] || any || 1 || We construct the characteristic polynomial of the differential equation, find its real and complex roots, and the space of solution functions is a vector space with basis functions described using these roots.
|}

===Qualitative methods===

For most differential equations, it is very hard to convert the differential equation to a series of integration problems and to find explicit expressions for the solution. Instead, in many cases, we try to determine the qualitative properties of solution functions, including existence, uniqueness, extent of differentiability, nature of roots and critical points, etc.

Proof of product rule for differentiation using chain rule for partial differentiation

2024-04-06T23:05:05Z

Vipul: Undo revision 3450 by Vipul (talk)

This article proves the [[product rule for differentiation]] in terms of the [[chain rule for partial differentiation]].

==Statements==

===Statement of product rule for differentiation (that we want to prove)===

uppose <math>f</math> and <math>g</math> are functions of one variable. Then the following is true wherever the right side expression makes sense (see [[concept of equality conditional to existence of one side]]):

<math>\! \frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)</math>

===Statement of chain rule for partial differentiation (that we want to use)===

Suppose <math>f,g</math> are both functions of one variable and <math>h</math> is a function of two variables. Suppose <math>u = f(x), v = g(x), w = h(u,v)</math>. Then: <math>\frac{dw}{dx} = \frac{\partial w}{\partial u}\frac{du}{dx} + \frac{\partial w}{\partial v} \frac{dv}{dx} = h_u(u,v)f'(x) + h_v(u,v)g'(x)</math>

<center>{{#widget:YouTube|id=3A78DVlZen8}}</center>

==Proof==

'''Given''': Functions <math>f</math> and <math>g</math>

'''To prove''': <math>\! \frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)</math> wherever the right side makes sense.

'''Proof''':

Consider the function:

<math>\! h(u,v) := uv</math>

Its partial derivatives are:

<math>\! h_u(u,v) = v, \qquad h_v(u,v) = u</math>
Define:

<math>\! u := f(x), \qquad v := g(x), \qquad w := uv = h(u,v) = f(x)g(x)</math>

By the chain rule for partial differentiation, we have:

<math>\! \frac{dw}{dx} = \frac{\partial w}{\partial u} \frac{du}{dx} + \frac{\partial w}{\partial v}\frac{dv}{dx}</math>

The left side is <math>(fg)'(x)</math>. The right side becomes:

<math>\! h_u(u,v)f'(x) + h_v(u,v)g'(x)</math>

This simplifies to:

<math>\! vf'(x) + ug'(x)</math>

Plug back the expressions <math>u = f(x), v = g(x)</math> and get:

<math>\! f'(x)g(x) + f(x)g'(x)</math>

Proof of product rule for differentiation using chain rule for partial differentiation

2024-04-06T23:02:59Z

Vipul:

This article proves the [[product rule for differentiation]] in terms of the [[chain rule for partial differentiation]].

==Statements==

===Statement of product rule for differentiation (that we want to prove)===

uppose <math>f</math> and <math>g</math> are functions of one variable. Then the following is true wherever the right side expression makes sense (see [[concept of equality conditional to existence of one side]]):

<math>\! \frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)</math>

===Statement of chain rule for partial differentiation (that we want to use)===

Suppose <math>f,g</math> are both functions of one variable and <math>h</math> is a function of two variables. Suppose <math>u = f(x), v = g(x), w = h(u,v)</math>. Then: <math>\frac{dw}{dx} = \frac{\partial w}{\partial u}\frac{du}{dx} + \frac{\partial w}{\partial v} \frac{dv}{dx} = h_u(u,v)f'(x) + h_v(u,v)g'(x)</math>

==Proof==

'''Given''': Functions <math>f</math> and <math>g</math>

'''To prove''': <math>\! \frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)</math> wherever the right side makes sense.

'''Proof''':

Consider the function:

<math>\! h(u,v) := uv</math>

Its partial derivatives are:

<math>\! h_u(u,v) = v, \qquad h_v(u,v) = u</math>
Define:

<math>\! u := f(x), \qquad v := g(x), \qquad w := uv = h(u,v) = f(x)g(x)</math>

By the chain rule for partial differentiation, we have:

<math>\! \frac{dw}{dx} = \frac{\partial w}{\partial u} \frac{du}{dx} + \frac{\partial w}{\partial v}\frac{dv}{dx}</math>

The left side is <math>(fg)'(x)</math>. The right side becomes:

<math>\! h_u(u,v)f'(x) + h_v(u,v)g'(x)</math>

This simplifies to:

<math>\! vf'(x) + ug'(x)</math>

Plug back the expressions <math>u = f(x), v = g(x)</math> and get:

<math>\! f'(x)g(x) + f(x)g'(x)</math>

User:Vipul

2024-04-06T22:59:20Z

Vipul:

User:Vipul

2024-04-06T22:38:07Z

Vipul:

Quiz:Equivalence of integration problems

2024-03-18T22:06:36Z

Vipul: /* General functions= */

This quiz considers questions about how one integration problem can be converted to another using [[integration by parts]] and [[integration by u-substitution]].

==General functions==

<quiz display=simple>
{Suppose <math>f</math> is a function with a known antiderivative <math>F</math>. Which of the following is correct (and can be deduced using integration by parts)?
|type="()"}
- Knowledge of an antiderivative for <math>x \mapsto f(x^2)</math> is equivalent to knowledge of an antiderivative for <math>F</math>.
- Knowledge of an antiderivative for <math>x \mapsto xf(x^2)</math> is equivalent to knowledge of an antiderivative for <math>F</math>.
- Knowledge of an antiderivative for <math>x \mapsto x^2f(x^2)</math> is equivalent to knowledge of an antiderivative for <math>F</math>.
- Knowledge of an antiderivative for <math>x \mapsto x^2f(x)</math> is equivalent to knowledge of an antiderivative for <math>F</math>.
+ Knowledge of an antiderivative for <math>x \mapsto xf(x)</math> is equivalent to knowledge of an antiderivative for <math>F</math>.

{Suppose <math>f</math> is a function with a known antiderivative <math>F</math>. Which of the following integration problems is ''not'' equivalent to the others?
|type="()"}
- <math>\int f(\sqrt{x}) \, dx</math>
- <math>\int xf(x) \, dx</math>
+ <math>\int f(x^2) \, dx</math>
- <math>\int F(x) \, dx</math>

{Suppose we know the first three antiderivatives for <math>f</math>, i.e., we have explicit expressions for an antiderivative of <math>f</math>, an antiderivative of that antiderivative, and an antiderivative of the antiderivative of the antiderivative. What is the largest nonnegative integer <math>k</math> for which this guarantees us an expression for an antiderivative of <math>x \mapsto x^kf(x)</math>?
|type="()"}
- 0
- 1
+ 2
- 3
- 4

{Suppose we know the first three antiderivatives for <math>f</math>, i.e., we have explicit expressions for an antiderivative of <math>f</math>, an antiderivative of that antiderivative, and an antiderivative of the antiderivative of the antiderivative. What is the largest nonnegative integer <math>k</math> for which this guarantees us an expression for an antiderivative of <math>x \mapsto f(x^{1/k})</math>? For simplicity, assume that we are only considering <math>x > 0</math>.
|type="()"}
- 0
- 1
- 2
+ 3
- 4
- 5

{Suppose <math>f</math> has a known antiderivative <math>F</math>. Consider the problems of integrating <math>f(x^2), xf(x^2), x^2f(x^2)</math>. What can we say about the relation between these problems?
|type="()"}
- All of these have antiderivatives expressible in terms of <math>F</math>.
- <math>f(x^2)</math> has an antiderivative expressible in terms of <math>F</math>. The integration problems for the other two functions are equivalent to each other.
+ <math>xf(x^2)</math> has an antiderivative expressible in terms of <math>F</math>. The integration problems for the other two functions are equivalent to each other.
- <math>x^2f(x^2)</math> has an antiderivative expressible in terms of <math>F</math>. The integration problems for the other two functions are equivalent to each other.
- All the integration problems are equivalent to each other, but none has a guaranteed expression in terms of <math>f</math> and <math>F</math>.

{Suppose <math>f</math> is an elementarily expressible and infinitely differentiable function on the positive reals (so all derivatives of <math>f</math> are also elementarily expressible). An antiderivative for <math>f''(x)/x</math> is '''not equivalent''' up to elementary functions to '''which one''' of the following?
|type="()"}
- An antiderivative for <math>x \mapsto f''(e^x)</math>, domain all of <math>\R</math>.
+ An antiderivative for <math>x \mapsto f'(e^x/x)</math>, domain positive reals.
- An antiderivative for <math>x \mapsto f'''(x)(\ln x)</math>, domain positive reals.
- An antiderivative for <math>x \mapsto f'(1/x)</math>, domain positive reals.
- An antiderivative for <math>x \mapsto f(1/\sqrt{x})</math>, domain positive reals.

</quiz>

==Specific functions==

<quiz display=simple>
{Suppose <math>a</math> and <math>b</math> are real numbers that are not positive integers. Which of the following is a ''sufficient'' condition for the integration problems <math>\int x^ae^x \, dx</math> and <math>\int x^be^x \, dx</math> to be equivalent?
|type="()"}
- <math>a + b</math> is an integer.
+ <math>a - b</math> is an integer.
|| For simplicity, assume <math>a < b</math> (the process works exactly the same way in reverse if <matH>b < a</math>). Start with the integral <math>\int x^be^x \, dx</math>. Now apply integration by parts taking <math>e^x</math> as the part to integrate and <math>x^b</math> as the part to differentiate. After one application of integration by parts, we need to integrate <math>x^{b-1}e^x</math>. Proceed in the way and we see that we get the integrations of <math>x^be^x, x^{b-1}e^x, x^{b-2}e^x, \dots</math>. If <math>a,b</math> differ by an integer, then after finitely many steps, we will land up with <math>\int x^a e^x\, dx</math>.
- <math>ab</math> is an integer.
- <math>a/b</math> is an integer.

{Suppose <math>a</math> and <math>b</math> are real numbers that are not positive integers. Which of the following is a ''sufficient'' condition for the integration problems <math>\int x^ae^x \, dx</math> and <math>\int e^{x^b} \, dx</math> to be equivalent? Assume we are working with <math>x > 0</math>, so any real power of <math>x</matH> makes sense.
|type="()"}
- <math>a + b = 1</math>
- <math>a - b = 1</math>
+ <math>ab = 1</math>
|| Using integration by parts once, we can convert <math>\int x^a e^x\, dx</math> to <math>\int ax^{a-1} e^x \, dx</math>. Now, put <math>u = x^a</math>. Then <math>x = u^{1/a}</math>, and <math>du = ax^{a-1} \, dx</math>. So, we get that the integral is <math>\int e^{u^{1/a}} \, du</math>. Replace the dummy variable <math>u</math> by the dummy variable <math>x</math>, to obtain <math>\int e^{x^{1/a}} \, dx</math>, which is <math>\int e^{x^b} \, dx</math> by the assumption that <math>b = 1/a</math>.
- <math>a/b = 1</math>

{Suppose <math>a</math> and <math>b</math> are positive real numbers. Which of the following is a ''sufficient'' condition for the integration problems <math>\int e^{x^a} \, dx</math> and <math>\int e^{x^b} \, dx</math> to be equivalent? Assume we are working with <math>x > 0</math>, so any real power of <math>x</matH> makes sense.
|type="()"}
- <math>1/a + 1/b</math> is an integer
+ <math>1/a - 1/b</math> is an integer
|| Put <math>u = x^a</math>. Then,we get <math>x = u^{1/a}</math> and the integral becomes <math>\int e^{x^a} \, dx= \frac{1}{a} \int e^u u^{1/a - 1} \, du</math>. If <matH>1/a - 1/b</math> is an integer, then repeated use of integration by parts gets us to <math>\int e^u u^{1/b - 1} \, du</math>. Now, we plug back <math>y = u^{1/b}</math> and get <math>\int e^{y^b} \, dy</math>. Constants are ignored here as they don't affect the equivalence of integration problems.
- <matH>1/(ab)</math> is an integer
- <math>a/b</math> is an integer

{Which of the following functions has an antiderivative that is '''not equivalent''' up to elementary functions to the antiderivative of <math>x \mapsto e^{-x^2}</math>?
+ <math>x \mapsto e^{-x^4}</math>
- <math>\mapsto e^{-x^{2/3}}</math>
|| Equivalent via <math>x \mapsto x^2e^{-x^2}</math>. Start with <math>\int e^{-x^{2/3}}</math>. Do a <math>u</math>-substitution <math>u = x^{1/3}</math>, get <matH>\int 3u^2e^{-u^2} \, du</math>.
- <math>x \mapsto e^{-x^{2/5}}</math>
|| Equivalent via <math>x \mapsto x^4e^{-x^2}</math>. Start with <math>\int e^{-x^{2/5}}</math>. Do a <math>u</math>-substitution <math>u = x^{1/5}</math>, get <matH>\int 5u^4e^{-u^2} \, du</math>.
- <math>x \mapsto x^2e^{-x^2}</math>
|| Consider <math>\int x^2e^{-x^2} \, dx</math>. Perform integration by parts on this, taking <math>xe^{-x^2} \, dx</math> as the part to integrate.
- <math>x \mapsto x^4e^{-x^2}</math>
|| Equivalent via <math>\int x^2e^{-x^2} \, dx</math>. Consider <math>\int x^4 e^{-x^2} \, dx</math>. Split as <matH>x^3 (xe^{-x^2})</math> and take <math>xe^{-x^2}</math> as the part to integrate, and in one step we get to <math>\int x^2e^{-x^2}</math>.

</quiz>

Quiz:Chain rule for differentiation

2024-03-18T21:40:11Z

Vipul: /* Formulas */

See [[chain rule for differentiation]] and [[chain rule for higher derivatives]] for background information.

See [[Quiz:Differentiation rules]] for a quiz on all the differentiation rules together.

==Formulas==

General difficulty level of questions in this section: College level (unless otherwise specified).

<quiz display=simple>
{Suppose <math>f</math> and <math>g</math> are both twice differentiable functions everywhere on <math>\R</math>. Which of the following is the correct formula for <math>(f \circ g)''</math>, the second derivative of the [[composite of two functions]]?
|type="()"}
- <math>(f'' \circ g) \cdot g''</math>
- <math>(f'' \circ g) \cdot (f' \circ g') \cdot g''</math>
- <math>(f'' \circ g) \cdot (f' \circ g') \cdot (f \circ g'')</math>
+ <math>(f'' \circ g) \cdot (g')^2 + (f' \circ g) \cdot g''</math>
|| See [[chain rule for differentiation]], [[chain rule for higher derivatives]].
- <math>(f' \circ g') \cdot (f \circ g) + (f'' \circ g'')</math>

{ Suppose <math>f_1,f_2,f_3</math> are everywhere differentiable functions from <math>\R</math> to <math>\R</math>. What is the derivative <math>(f_1 \circ f_2 \circ f_3)'</math> where <math>\circ</math> denotes the [[composite of two functions]]? In other words, <math>(f_1 \circ f_2 \circ f_3)(x) := f_1(f_2(f_3(x)))</math>.
|type="()"}
+ <math>(f_1' \circ f_2 \circ f_3) \cdot (f_2' \circ f_3) \cdot f_3'</math>
|| See [[chain rule for differentiation#Statement for multiple functions]]
- <math>(f_1' \cdot f_2 \cdot f_3) \circ (f_2' \cdot f_3) \circ f_3'</math>
- <math>(f_1 \circ f_2' \circ f_3') \cdot (f_2 \circ f_3') \cdot f_3</math>
- <math>(f_1 \cdot f_2' \cdot f_3') \circ (f_2 \cdot f_3') \circ f_3</math>
- <math>f_1' \circ f_2' \circ f_3'</math>

{ Suppose <math>f</math> is a differentiable function from <math>\R</math> to <math>\R</math> and <math>a, b \in \R</math> are such that <math>f(a) = a</math> and <math>f'(a) = b</math>. What is the value of <math>(f \circ f \circ \dots \circ f)'(a)</math>, where <math>\circ</math> denotes the [[composite of two functions]] and <math>f</math> occurs <math>n</math> times in the expression, with <math>n \ge 3</math>?
|type="()"}
- <math>a^n</math>
- <math>a^{n - 1}b</math>
- <math>a^{n - 1}b + ab^{n - 1}</math>
- <math>ab^{n - 1}</math>
+ <math>b^n</math>
|| The chain rule gives the derivative as a product of <math>n</math> terms, each of which is of the form <math>f'</math> applied to <math>k</math> iterates of <math>f</math>, with <math>k</math> varying from <math>0</math> to <math>n - 1</math>. Evaluating at <math>a</math> and using <math>f'(a) = a</math>, each term simplifies to <math>f'(a)</math> and hence to <math>b</math>. As there are <math>n</math> such terms, the product is <math>b^n</math>. Note that <math>n \ge 3</math> is not necessary (this reasoning works for <math>n = 1</math> and <math>n = 2</math> as well). That condition was added primarily to dissuade people from using <math>n = 1</math> or <math>n = 2</math> to figure out the correct answer by a process of elimination.

</quiz>

Quiz:Chain rule for differentiation

2024-03-18T21:39:13Z

Vipul: /* Formulas */

See [[chain rule for differentiation]] and [[chain rule for higher derivatives]] for background information.

See [[Quiz:Differentiation rules]] for a quiz on all the differentiation rules together.

==Formulas==

<quiz display=simple>
{Suppose <math>f</math> and <math>g</math> are both twice differentiable functions everywhere on <math>\R</math>. Which of the following is the correct formula for <math>(f \circ g)''</math>, the second derivative of the [[composite of two functions]]?
|type="()"}
- <math>(f'' \circ g) \cdot g''</math>
- <math>(f'' \circ g) \cdot (f' \circ g') \cdot g''</math>
- <math>(f'' \circ g) \cdot (f' \circ g') \cdot (f \circ g'')</math>
+ <math>(f'' \circ g) \cdot (g')^2 + (f' \circ g) \cdot g''</math>
|| See [[chain rule for differentiation]], [[chain rule for higher derivatives]].
- <math>(f' \circ g') \cdot (f \circ g) + (f'' \circ g'')</math>

{ Suppose <math>f_1,f_2,f_3</math> are everywhere differentiable functions from <math>\R</math> to <math>\R</math>. What is the derivative <math>(f_1 \circ f_2 \circ f_3)'</math> where <math>\circ</math> denotes the [[composite of two functions]]? In other words, <math>(f_1 \circ f_2 \circ f_3)(x) := f_1(f_2(f_3(x)))</math>.
|type="()"}
+ <math>(f_1' \circ f_2 \circ f_3) \cdot (f_2' \circ f_3) \cdot f_3'</math>
|| See [[chain rule for differentiation#Statement for multiple functions]]
- <math>(f_1' \cdot f_2 \cdot f_3) \circ (f_2' \cdot f_3) \circ f_3'</math>
- <math>(f_1 \circ f_2' \circ f_3') \cdot (f_2 \circ f_3') \cdot f_3</math>
- <math>(f_1 \cdot f_2' \cdot f_3') \circ (f_2 \cdot f_3') \circ f_3</math>
- <math>f_1' \circ f_2' \circ f_3'</math>

{ Suppose <math>f</math> is a differentiable function from <math>\R</math> to <math>\R</math> and <math>a, b \in \R</math> are such that <math>f(a) = a</math> and <math>f'(a) = b</math>. What is the value of <math>(f \circ f \circ \dots \circ f)'(a)</math>, where <math>\circ</math> denotes the [[composite of two functions]] and <math>f</math> occurs <math>n</math> times in the expression, with <math>n \ge 3</math>?
|type="()"}
- <math>a^n</math>
- <math>a^{n - 1}b</math>
- <math>a^{n - 1}b + ab^{n - 1}</math>
- <math>ab^{n - 1}</math>
+ <math>b^n</math>
|| The chain rule gives the derivative as a product of <math>n</math> terms, each of which is of the form <math>f'</math> applied to <math>k</math> iterates of <math>f</math>, with <math>k</math> varying from <math>0</math> to <math>n - 1</math>. Evaluating at <math>a</math> and using <math>f'(a) = a</math>, each term simplifies to <math>f'(a)</math> and hence to <math>b</math>. As there are <math>n</math> such terms, the product is <math>b^n</math>. Note that <math>n \ge 3</math> is not necessary (this reasoning works for <math>n = 1</math> and <math>n = 2</math> as well). That condition was added primarily to dissuade people from using <math>n = 1</math> or <math>n = 2</math> to figure out the correct answer by a process of elimination.

</quiz>