Difference between revisions of "Second derivative"
(Created page with "==Definition at a point== The '''second derivative''' of a function <math>f</math> at a point <math>x_0</math>, denoted <math>f''(x_0)</math>, is defined as the derivative a...") |
(→Significance) |
||
(4 intermediate revisions by the same user not shown) | |||
Line 1: | Line 1: | ||
==Definition at a point== | ==Definition at a point== | ||
− | The '''second derivative''' of a function <math>f</math> at a point <math>x_0</math>, denoted <math>f''(x_0)</math>, is defined as the [[derivative]] at the point <math>x_0</math> of the function defined as the [[derivative]] <math>f'</math> | + | ===Definition in terms of first derivative=== |
+ | |||
+ | The '''second derivative''' of a function <math>f</math> at a point <math>x_0</math>, denoted <math>\! f''(x_0)</math>, is defined as the [[derivative]] at the point <math>x_0</math> of the function defined as the [[derivative]] <math>f'</math> | ||
Note that the ''first'' differentiation operation must be performed, not just at the point, but at all points near it, so that we have a ''function'' for the first derivative around the point, which we can then differentiate to calculate the second derivative ''at'' the point. It is ''not'' good enough to calculate the first derivative only ''at'' the particular point (i.e., to calculate ''only'' <math>f'(x_0)</math>) and then proceed to differentiate that; we need the value of the first derivative at nearby points too. | Note that the ''first'' differentiation operation must be performed, not just at the point, but at all points near it, so that we have a ''function'' for the first derivative around the point, which we can then differentiate to calculate the second derivative ''at'' the point. It is ''not'' good enough to calculate the first derivative only ''at'' the particular point (i.e., to calculate ''only'' <math>f'(x_0)</math>) and then proceed to differentiate that; we need the value of the first derivative at nearby points too. | ||
+ | |||
+ | ===Definition as a limit expression=== | ||
+ | |||
+ | The '''second derivative''' of a function <math>f</math> at a point <math>x_0</math>, denoted <math>\! f''(x_0)</math>, is defined as follows: | ||
+ | |||
+ | <math>\! \lim_{x \to x_0} \frac{f'(x) - f'(x_0)}{x - x_0}</math> | ||
+ | |||
+ | More explicitly, this can be written as: | ||
+ | |||
+ | <math>\! \lim_{x \to x_0} \frac{1}{x - x_0}\left[\lim_{x_1 \to x} \frac{f(x_1) - f(x)}{x_1 - x} - \lim_{x_2 \to x_0} \frac{f(x_2) - f(x_0)}{x_2 - x_0}\right]</math> | ||
==Definition as a function== | ==Definition as a function== | ||
Line 10: | Line 22: | ||
<math>\! f'' := (f')'</math> | <math>\! f'' := (f')'</math> | ||
+ | |||
+ | ==Leibniz notation for second 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{d^2y}{dx^2} := f''(x)</math> | ||
+ | |||
+ | This can also be written as: | ||
+ | |||
+ | <math>\frac{d^2}{dx^2}[f(x)]</math> | ||
+ | |||
+ | In particular, <math>d^2y/dx^2</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{d^2y}{dx^2} |_{x = x_0} := f''(x_0)</math> | ||
+ | |||
+ | ==Significance== | ||
+ | |||
+ | ===Significance of sign on intervals=== | ||
+ | |||
+ | {| class="sortable" border="1" | ||
+ | ! Loose statement !! Precise statements | ||
+ | |- | ||
+ | | [[concave up function]] means derivative is increasing means second derivative is positive || [[positive second derivative implies concave up]]<br>{{fillin}} | ||
+ | |- | ||
+ | | [[concave down function]] means derivative is decreasing means second derivative is negative || [[negative second derivative implies concave down]] | ||
+ | |- | ||
+ | | [[linear function]] (or [[constant function]]) means derivative is constant means second derivative is zero || | ||
+ | |} | ||
+ | |||
+ | ===Significance of sign at points=== | ||
+ | |||
+ | A [[point of inflection]] is a point of geometric significance on the graph (where the function changes its sense of concavity), and corresponds to a point in the domain of the function where the second derivative changes sign. A point of inflection must correspond to a point of (strict) local maximum or minimum for the [[first derivative]]. | ||
+ | |||
+ | {| class="sortable" border="1" | ||
+ | ! Method for constructing new functions from old !! In symbols !! Derivative in terms of old functions and their first, second derivatives !! Proof | ||
+ | |- | ||
+ | | [[pointwise sum of functions|pointwise sum]] || <math>f + g</math> is the function <math>x \mapsto f(x) + g(x)</math><br><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 second derivatives of the functions being added (''the second derivative of the sum is the sum of the second derivatives'')<br><math>\! f'' + g''</math><br><math>\! f_1'' + f_2'' + \dots + f_n''</math> || [[repeated 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 second derivatives, i.e., <math>f'' - g''</math> || [[repeated 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> || [[repeated 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><br><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) + 2f'(x)g'(x) + f(x)g''(x)</math><br>For multiple functions, more complicated || [[product rule for higher derivatives]] | ||
+ | |- | ||
+ | | [[pointwise quotient of functions|pointwise quotient]] || <math>f/g</math> is the function <math>x \mapsto f(x)/g(x)</math> || {{fillin}} || [[quotient rule for higher derivatives]] | ||
+ | |- | ||
+ | | [[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))^2 + f'(g(x))g''(x)</math> || [[chain rule for higher derivatives]] | ||
+ | |- | ||
+ | | [[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{-f''(f^{-1}(x))}{(f'(f^{-1}(x)))^3}</math> || [[higher derivatives of inverse function]] | ||
+ | |- | ||
+ | | [[piecewise definition of functions|piecewise definition]] || {{fillin}} || {{fillin}} || [[differentiation rule for piecewise definition by interval]] | ||
+ | |} |
Latest revision as of 03:15, 18 December 2011
Contents
Definition at a point
Definition in terms of first derivative
The second derivative of a function at a point , denoted , is defined as the derivative at the point of the function defined as the derivative
Note that the first differentiation operation must be performed, not just at the point, but at all points near it, so that we have a function for the first derivative around the point, which we can then differentiate to calculate the second derivative at the point. It is not good enough to calculate the first derivative only at the particular point (i.e., to calculate only ) and then proceed to differentiate that; we need the value of the first derivative at nearby points too.
Definition as a limit expression
The second derivative of a function at a point , denoted , is defined as follows:
More explicitly, this can be written as:
Definition as a function
The second derivative of a function at a point is defined as the derivative of the derivative of the function. For a function , the second derivative is defined as:
Leibniz notation for second derivative
Suppose is a function, and are variables related by . Here, is an independent variable and is the dependent variable (with the dependency being described by the function ). We then define:
This can also be written as:
In particular, is a function of . Its value at is defined as and is denoted as follows:
Significance
Significance of sign on intervals
Loose statement | Precise statements |
---|---|
concave up function means derivative is increasing means second derivative is positive | positive second derivative implies concave up Fill this in later |
concave down function means derivative is decreasing means second derivative is negative | negative second derivative implies concave down |
linear function (or constant function) means derivative is constant means second derivative is zero |
Significance of sign at points
A point of inflection is a point of geometric significance on the graph (where the function changes its sense of concavity), and corresponds to a point in the domain of the function where the second derivative changes sign. A point of inflection must correspond to a point of (strict) local maximum or minimum for the first derivative.
Method for constructing new functions from old | In symbols | Derivative in terms of old functions and their first, second derivatives | Proof |
---|---|---|---|
pointwise sum | is the function is the function |
Sum of the second derivatives of the functions being added (the second derivative of the sum is the sum of the second derivatives) |
repeated differentiation is linear |
pointwise difference | is the function | Difference of the second derivatives, i.e., | repeated differentiation is linear |
scalar multiple by a constant | is the function where is a real number | repeated differentiation is linear | |
pointwise product | (sometimes denoted ) is the function (sometimes denoted is the function |
For two functions, For multiple functions, more complicated |
product rule for higher derivatives |
pointwise quotient | is the function | Fill this in later | quotient rule for higher derivatives |
composite of two functions | is the function | chain rule for higher derivatives | |
inverse function of a one-one function | sends to the unique such that | higher derivatives of inverse function | |
piecewise definition | Fill this in later | Fill this in later | differentiation rule for piecewise definition by interval |