Difference between revisions of "Partial derivative"
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+  {{multivariable analogue ofderivative}}  
+  
==Definition at a point==  ==Definition at a point==  
===Generic definition===  ===Generic definition===  
−  Suppose <math>f</math> is a function of more than one variable, where <math>x</math> is one of the input variables to <math>f</math>. Fix a choice <math>x = x_0</math> and fix the values of all the other variables. The '''partial derivative''' of <math>f</math> with respect to <math>x</math>, denoted <math>\partial f/\partial x</math>, or <math>f_x</math>, is defined as the derivative at <math>x_0</math> of the function that sends <math>x</math> to <math>f</math> at <math>x</math> for the same fixed choice of the other input variables.  +  Suppose <math>f</math> is a function of more than one variable, where <math>x</math> is one of the input variables to <math>f</math>. Fix a choice <math>x = x_0</math> and fix the values of all the other variables. The '''partial derivative''' of <math>f</math> with respect to <math>x</math> ''at the point'', denoted <math>\partial f/\partial x</math>, or <math>f_x</math>, is defined as the derivative at <math>x_0</math> of the function that sends <math>x</math> to <math>f</math> at <math>x</math> for the same fixed choice of the other input variables. 
+  
+  <center>{{#widget:YouTubeid=rsZc1SYuiig}}</center>  
===For a function of two variables===  ===For a function of two variables===  
−  Suppose <math>f</math> is a realvalued function of two variables <math>x,y</math>, i.e., the [[domain]] of <math>f</math> is a subset of <math>\R^2</math>. We define the partial derivatives as follows:  +  Suppose <math>f</math> is a realvalued function of two variables <math>x,y</math>, i.e., the [[domain]] of <math>f</math> is a subset of <math>\R^2</math>. Suppose <math>(x_0,y_0)</math> is a point in the [[domain]] of <math>f</math>, i.e., it's the point with <math>x = x_0</math> and <math>y = y_0</math> (here, <math>x_0,y_0</math> are actual numerical values). We define the partial derivatives at <math>(x_0,y_0)</math> as follows: 
+  
+  { class="sortable" border="1"  
+  ! Item !! For partial derivative with respect to <math>x</math> !! For partial derivative with respect to <math>y</math>  
+    
+   Notation  <math>\frac{\partial f(x,y)}{\partial x}_{(x,y) = (x_0,y_0)}</math><br>Also denoted <math>f_x(x_0,y_0)</math> or <math>f_1(x_0,y_0)</math>  <math>\frac{\partial f(x,y)}{\partial y}_{(x,y) = (x_0,y_0)}</math><br>Also denoted <math>f_y(x_0,y_0)</math> or <math>f_2(x_0,y_0)</math>  
+    
+   Definition as [[derivative]]  <math>\frac{d}{dx}f(x,y_0)_{x = x_0}</math>. In other words, it is the derivative (at <math>x = x_0</math>) of the function <math>x \mapsto f(x,y_0)</math>  <math>\frac{d}{dy}f(x_0,y)_{y = y_0}</math>. In other words, it is the derivative (at <math>y = y_0</math>) of the function <math>y \mapsto f(x_0,y)</math>.  
+    
+   Definition as limit (using derivative as limit of difference quotient)  <math>\lim_{x \to x_0} \frac{f(x,y_0)  f(x_0,y_0)}{x  x_0}</math><br><math>\lim_{h \to 0} \frac{f(x_0 + h,y_0)  f(x_0,y_0)}{h}</math>  <math>\lim_{y \to y_0} \frac{f(x_0,y)  f(x_0,y_0)}{y  y_0}</math><br><math>\lim_{h \to 0} \frac{f(x_0,y_0 + h)  f(x_0,y_0)}{h}</math>  
+    
+   Definition as [[directional derivative]]  Directional derivative at <math>(x_0,y_0)</math> with respect to a unit vector in the positive <math>x</math>direction.  Directional derivative at <math>(x_0,y_0)</math> with respect to a unit vector in the positive <math>y</math>direction.  
+  }  
+  
+  <center>{{#widget:YouTubeid=P9ciiweXh_A}}{{#widget:YouTubeid=j6TthcrlvoU}}</center>  
−  +  ===For a function of multiple variables===  
−  <math>\frac{\partial f(  +  The notation here gets a little messy, so read it carefully. We consider a function <math>f</math> of <math>n</math> variables, which we generically denote <math>(x_1,x_2,\dots,x_n)</math> respectively. Consider a point <math>(a_1,a_2,\dots,a_n)</math> in the domain of the function. In other words, this is a point where <math>x_1 = a_1,x_2 =a_2, \dots, x_n = a_n</math>. 
+  
+  Suppose <math>i</math> is a natural number in the set <math>\{ 1,2,3,\dots,n \}</math>.  
+  
+  { class="sortable" border="1"  
+  ! Item !! Value for partial derivative with respect to <math>x_i</math>  
+    
+   Notation  <math>\frac{\partial}{\partial x_i}f(x_1,x_2,\dots,x_n)_{(x_1,x_2,\dots,x_n) = (a_1,a_2,\dots,a_n)}</math><br>Also denoted <math>f_{x_i}(a_1,a_2,\dots,a_n)</math> or <math>f_i(a_1,a_2,\dots,a_n)</math>  
+    
+   Definition as [[derivative]]  <math>\frac{d}{dx_i}f(a_1,a_2,\dots,a_{i1},x_i,a_{i+1}, \dots,a_n)_{x_i = a_i}</math>. In other words, it is the derivative of the function <math>x_i \mapsto f(a_1,a_2,\dots,a_{i1},x_i,a_{i+1},\dots,a_n)</math> with respect to <math>x_i</math>, evaluated at the point <math>x_i = a_i</math>.  
+    
+   Definition as a [[limit]] (using derivative as limit of difference quotient)  <math>\lim_{x_i \to a_i} \frac{f(a_1,a_2,\dots,a_{i1},x_i,a_{i+1},\dots,a_n)  f(a_1,a_2,\dots,a_n)}{x_i  a_i}</math>  
+    
+   Definition as a [[directional derivative]]  Directional derivative in the positive <math>x_i</math>direction.  
+  }  
+  
+  <center>{{#widget:YouTubeid=bWnXXfMleE}}</center>  
+  
+  ==Definition as a function==  
+  
+  ===Generic definition===  
+  Suppose <math>f</math> is a function of more than one variable, where <math>x</math> is one of the input variables to <math>f</math>. The '''partial derivative''' of <math>f</math> with respect to <math>x</math>, denoted <math>\partial f/\partial x</math>, or <math>f_x</math> is defined as the ''function'' that sends points in the domain of <math>f</math> (including values of all the variables) to the partial derivative with respect to <math>x</math> of <math>f</math> (i.e., the derivative treating the other inputs as constants for the computation of the derivative). In particular, the domain of the partial derivative of <math>f</math> with respect to <math>x</math> is a subset of the domain of <math>f</math>.  
−  +  We can compute the partial derivative of <math>f</matH> relative to each of the inputs to <math>f</math>.  
−  +  {{generic point specific point confusion}}  
−  +  ===For a function of two variables===  
−  <math>  +  Suppose <math>f</math> is a realvalued function of two variables <math>x,y</math>, i.e., the [[domain]] of <math>f</math> is a subset of <math>\R^2</math>. The partial derivatives of <math>f</math> with respect to <math>x</math> and <math>y</math> are ''both'' functions of two variables each of which has domain a subset of the domain of <math>f</math>. 
−  +  { class="sortable" border="1"  
+  ! Item !! For partial derivative with respect to <math>x</math> !! For partial derivative with respect to <math>y</math>  
+    
+   Notation  <math>\frac{\partial f(x,y)}{\partial x}</math><br>Also denoted <math>f_x(x,y)</math> or <math>f_1(x,y)</math>  <math>\frac{\partial f(x,y)}{\partial y}</math><br>Also denoted <math>f_y(x,y)</math> or <math>f_2(x,y)</math>  
+    
+   Definition as [[derivative]]  It is the derivative of the function <math>x \mapsto f(x,y)</math>, treating <math>y</math> as an unknown constant  It is the derivative of the function <math>y \mapsto f(x,y)</math>, treating <math>x</math> as an unknown constant  
+    
+   Definition as limit (using derivative as limit of difference quotient)  <math>\lim_{h \to 0} \frac{f(x + h,y)  f(x,y)}{h}</math>  <math>\lim_{h \to 0} \frac{f(x,y + h)  f(x,y)}{h}</math>  
+    
+   Definition as [[directional derivative]]  Directional derivative with respect to a unit vector in the positive <math>x</math>direction.  Directional derivative with respect to a unit vector in the positive <math>y</math>direction.  
+  }  
−  +  <center>{{#widget:YouTubeid=0zu9BstjjZA}}</center>  
===For a function of multiple variables===  ===For a function of multiple variables===  
−  {{  +  { class="sortable" border="1" 
+  ! Item !! Value for partial derivative with respect to <math>x_i</math>  
+    
+   Notation  <math>\frac{\partial}{\partial x_i}f(x_1,x_2,\dots,x_n)</math><br>Also denoted <math>f_{x_i}(x_1,x_2,\dots,x_n)</math> or <math>f_i(x_1,x_2,\dots,x_n)</math>  
+    
+   Definition as [[derivative]]  It is the derivative of the function <math>x_i \mapsto f(x_1,x_2,\dots,x_{i1},x_i,x_{i+1},\dots,x_n)</math> with respect to <math>x_i</math>, where all the other variables are treated as unknown constants while doing the differentiation.  
+    
+   Definition as a [[limit]] (using derivative as limit of difference quotient)  <math>\lim_{h \to 0} \frac{f(x_1,x_2,\dots,x_{i1},x_i + h, x_{i+1},\dots,x_n)  f(x_1,x_2,\dots,x_n)}{h}</math>  
+    
+   Definition as a [[directional derivative]]  Directional derivative in the positive <math>x_i</math>direction.  
+  }  
+  
+  ==Graphical interpretation==  
+  
+  ===For a function of two variables at a point===  
+  
+  Suppose <math>f</math> is a function of two variables <math>x,y</math> and <math>(x_0,y_0)</math> is a point in the domain of the function. Consider the graph of <math>f</math> in threedimensional space, given by <math>z =f(x,y)</math>.  
+  
+  We have the following:  
+  
+  { class="sortable" border="1"  
+  ! Partial derivative !! Graphical interpretation  
+    
+   The partial derivative <math>f_x(x_0,y_0)</math> at a point <math>(x_0,y_0)</math> in the domain of the function  The slope of the tangent line at <math>x= x_0</math> to the restriction of the graph of <math>f</math> to the plane <math>y = y_0</math>.  
+    
+   The partial derivative <math>f_y(x_0,y_0)</math> at a point <math>(x_0,y_0)</math> in the domain of the function  The slope of the tangent line at <math>y = y_0</math> to the restriction of the graph of <math>f</math> to the plane <math>x = x_0</math>.  
+  }  
+  
+  <center>{{#widget:YouTubeid=yMuZIoGjTo}}</center>  
+  
+  ===For a function of multiple variables at a point===  
+  
+  Suppose <math>f</math> is a function of <math>n</math> variables <math>x_1,x_2,\dots,x_n</math> and suppose <math>(a_1,a_2,\dots,a_n)</math> is a point in the domain of <math>f</math>. Consider the graph of <math>f</math> in <math>\R^{n+1}</matH> given by:  
+  
+  <math>x_{n+1} = f(x_1,x_2,\dots,x_n)</math>  
+  
+  For any <math>i \in \{ 1,2,\dots,n\}</math>, we define the partial derivative <math>f_{x_i}(a_1,a_2,\dots,a_n)</math>, also denoted <math>f_i(a_1,a_2,\dots,a_n)</math>, as follows:  
+  
+  * First, consider the intersection of the graph of <math>f</math> with the plane given by the set of <math>n  1</math> equations <math>x_j = a_j</math> for all <math>j \ne i</math>. This is a plane parallel to the <math>x_ix_{n+1}</math>plane.  
+  * In this plane, consider the slope of the tangent line at <math>x_i =a_i</math>. This is the value of the partial derivative.  
+  
+  ==Related notions==  
+  
+  * [[Derivative]]  
+  * [[Higher derivative]]  
+  * [[Higher partial derivative]]  
+  
+  ==Domain considerations==  
+  
+  As already noted in the definition of partial derivative, the domain of the partial derivative of a function with respect to a variable is a subset of the domain of the function. However, we can actually say a little more.  
+  
+  ===For a function of two variables===  
+  
+  Suppose <math>f</math> is a function of two variables <math>x,y</math>. Then, a ''necessary'' condition for us to make sense of the partial derivative <math>f_x</math> at a point <math>(x_0,y_0)</math> is that <math>f</math> be defined on a small open interval about the point <math>x_0</math> on the line <matH>y = y_0</math>. Note that it is ''not necessary'' that <math>f</math> actually be defined in an open ball surrounding the point <math>(x_0,y_0)</math>  the only thing that matters is that <math>f</math> be defined under slight perturbations of <math>x</math>, holding <math>y</math> constant.  
+  
+  Similar remarks apply to <math>f_y</math>: a ''necessary'' condition for us to make sense of the partial derivative <math>f_y</math> at a point <math>(x_0,y_0)</math> is that <math>f</math> be defined on a small open interval about the point <math>y_0</math> on the line <matH>x = x_0</math>.  
+  
+  Consider, for instance, a function defined on the set <math>[0,1] \times [0,1]</math>, i.e., the set <math>\{ (x,y) \mid 0 \le x \le 1, 0 \le y \le 1 \}</math>. It makes sense to try computing the partial derivative <math>f_x</math> at all points in the subset <math>(0,1) \times [0,1]</math>, i.e., all points whose <math>x</math>coordinate is ''strictly'' between <math>0</math> and <matH>1</math>, but the <math>y</matH>coordinate is allowed to take the extreme values 0 and 1. Similarly, it makes sense to try computing the partial derivative <math>f_y</math> at all points in the subset <math>[0,1] \times (0,1)</math>, i.e., all points whose <math>y</math>coordinate is ''strictly'' between <math>0</math> and <matH>1</math>, but the <math>x</matH>coordinate is allowed to take the extreme values 0 and 1.  
+  
+  Note that the above only refers to the points at which it makes sense to ''try'' computing the partial derivative. It may still turn out that the partial derivative does not exist at many of these points.  
+  
+  <center>{{#widget:YouTubeid=9Q9OrXye748}}</center>  
+  
+  ==Caveats==  
+  
+  ===Value of partial derivative depends on all inputs===  
+  
+  {{further[[Value of partial derivative depends on all inputs]]}}  
+  
+  {{#lst:value of partial derivative depends on all inputsexample 1}}  
+  
+  ===Meaning of partial derivative depends on entire coordinate system===  
+  
+  {{further[[Meaning of partial derivative depends on entire coordinate system]]}}  
+  
+  This is a very subtle but very important point. It says that the ''partial derivative'' with respect to one variable depends not only on the choice of that particular variable, but on the choice of the other variables that are being kept constant for the purpose of computing the partial derivative. If a coordinate transformation is performed that changes what those ''other variables'' are, that could affect the value of the partial derivative.  
+  
+  This has a very realworld corollary. In economics and social science, we often talk of the partial derivative with respect to one variable as measuring what happens ''ceteris paribus'' on the other variables. However, the notion of ''ceteris paribus'' on other variables depends on what the other variables are. If we redefine the coordinate system to change that meaning, the partial derivative can change.  
+  
+  {{#lst:meaning of partial derivative depends on entire coordinate systemexample 1}} 
Latest revision as of 16:58, 25 January 2013
This article describes an analogue for functions of multiple variables of the following term/fact/notion for functions of one variable: derivative
Contents
Definition at a point
Generic definition
Suppose is a function of more than one variable, where is one of the input variables to . Fix a choice and fix the values of all the other variables. The partial derivative of with respect to at the point, denoted , or , is defined as the derivative at of the function that sends to at for the same fixed choice of the other input variables.
For a function of two variables
Suppose is a realvalued function of two variables , i.e., the domain of is a subset of . Suppose is a point in the domain of , i.e., it's the point with and (here, are actual numerical values). We define the partial derivatives at as follows:
Item  For partial derivative with respect to  For partial derivative with respect to 

Notation  Also denoted or 
Also denoted or 
Definition as derivative  . In other words, it is the derivative (at ) of the function  . In other words, it is the derivative (at ) of the function . 
Definition as limit (using derivative as limit of difference quotient)  

Definition as directional derivative  Directional derivative at with respect to a unit vector in the positive direction.  Directional derivative at with respect to a unit vector in the positive direction. 
For a function of multiple variables
The notation here gets a little messy, so read it carefully. We consider a function of variables, which we generically denote respectively. Consider a point in the domain of the function. In other words, this is a point where .
Suppose is a natural number in the set .
Item  Value for partial derivative with respect to 

Notation  Also denoted or 
Definition as derivative  . In other words, it is the derivative of the function with respect to , evaluated at the point . 
Definition as a limit (using derivative as limit of difference quotient)  
Definition as a directional derivative  Directional derivative in the positive direction. 
Definition as a function
Generic definition
Suppose is a function of more than one variable, where is one of the input variables to . The partial derivative of with respect to , denoted , or is defined as the function that sends points in the domain of (including values of all the variables) to the partial derivative with respect to of (i.e., the derivative treating the other inputs as constants for the computation of the derivative). In particular, the domain of the partial derivative of with respect to is a subset of the domain of .
We can compute the partial derivative of relative to each of the inputs to .
MORE ON THE WAY THIS DEFINITION OR FACT IS PRESENTED: We first present the version that deals with a specific point (typically with a subscript) in the domain of the relevant functions, and then discuss the version that deals with a point that is free to move in the domain, by dropping the subscript. Why do we do this?
The purpose of the specific point version is to emphasize that the point is fixed for the duration of the definition, i.e., it does not move around while we are defining the construct or applying the fact. However, the definition or fact applies not just for a single point but for all points satisfying certain criteria, and thus we can get further interesting perspectives on it by varying the point we are considering. This is the purpose of the second, generic point version.
For a function of two variables
Suppose is a realvalued function of two variables , i.e., the domain of is a subset of . The partial derivatives of with respect to and are both functions of two variables each of which has domain a subset of the domain of .
Item  For partial derivative with respect to  For partial derivative with respect to 

Notation  Also denoted or 
Also denoted or 
Definition as derivative  It is the derivative of the function , treating as an unknown constant  It is the derivative of the function , treating as an unknown constant 
Definition as limit (using derivative as limit of difference quotient)  
Definition as directional derivative  Directional derivative with respect to a unit vector in the positive direction.  Directional derivative with respect to a unit vector in the positive direction. 
For a function of multiple variables
Item  Value for partial derivative with respect to 

Notation  Also denoted or 
Definition as derivative  It is the derivative of the function with respect to , where all the other variables are treated as unknown constants while doing the differentiation. 
Definition as a limit (using derivative as limit of difference quotient)  
Definition as a directional derivative  Directional derivative in the positive direction. 
Graphical interpretation
For a function of two variables at a point
Suppose is a function of two variables and is a point in the domain of the function. Consider the graph of in threedimensional space, given by .
We have the following:
Partial derivative  Graphical interpretation 

The partial derivative at a point in the domain of the function  The slope of the tangent line at to the restriction of the graph of to the plane . 
The partial derivative at a point in the domain of the function  The slope of the tangent line at to the restriction of the graph of to the plane . 
For a function of multiple variables at a point
Suppose is a function of variables and suppose is a point in the domain of . Consider the graph of in given by:
For any , we define the partial derivative , also denoted , as follows:
 First, consider the intersection of the graph of with the plane given by the set of equations for all . This is a plane parallel to the plane.
 In this plane, consider the slope of the tangent line at . This is the value of the partial derivative.
Related notions
Domain considerations
As already noted in the definition of partial derivative, the domain of the partial derivative of a function with respect to a variable is a subset of the domain of the function. However, we can actually say a little more.
For a function of two variables
Suppose is a function of two variables . Then, a necessary condition for us to make sense of the partial derivative at a point is that be defined on a small open interval about the point on the line . Note that it is not necessary that actually be defined in an open ball surrounding the point  the only thing that matters is that be defined under slight perturbations of , holding constant.
Similar remarks apply to : a necessary condition for us to make sense of the partial derivative at a point is that be defined on a small open interval about the point on the line .
Consider, for instance, a function defined on the set , i.e., the set . It makes sense to try computing the partial derivative at all points in the subset , i.e., all points whose coordinate is strictly between and , but the coordinate is allowed to take the extreme values 0 and 1. Similarly, it makes sense to try computing the partial derivative at all points in the subset , i.e., all points whose coordinate is strictly between and , but the coordinate is allowed to take the extreme values 0 and 1.
Note that the above only refers to the points at which it makes sense to try computing the partial derivative. It may still turn out that the partial derivative does not exist at many of these points.
Caveats
Value of partial derivative depends on all inputs
For further information, refer: Value of partial derivative depends on all inputs
For instance, consider:
Then, we have:
and:
Note that each of the expressions involves both the variables and . In particular, this means that the value of at a point depends on both the coordinate and the coordinate of the point. Thus, for instance:
Despite the same value of 2 in both cases, the values are different because of differences in the input values.
Similarly, consider:
Despite the same value of 4 in both cases, the values are different because of differences in the input values.
Meaning of partial derivative depends on entire coordinate system
For further information, refer: Meaning of partial derivative depends on entire coordinate system
This is a very subtle but very important point. It says that the partial derivative with respect to one variable depends not only on the choice of that particular variable, but on the choice of the other variables that are being kept constant for the purpose of computing the partial derivative. If a coordinate transformation is performed that changes what those other variables are, that could affect the value of the partial derivative.
This has a very realworld corollary. In economics and social science, we often talk of the partial derivative with respect to one variable as measuring what happens ceteris paribus on the other variables. However, the notion of ceteris paribus on other variables depends on what the other variables are. If we redefine the coordinate system to change that meaning, the partial derivative can change.
Consider the function:
In this case, we have:
Now, suppose we consider in terms of and . Then, we have . Rewriting in terms of and , we get:
In other words, we can define as a function of two variables and . If we use the letter to denote this new function, we get:
In this case, we have:
Note that the two partial derivatives with respect to are not equal. The reason for this is that in the first case, we are taking the partial derivative with respect to keeping constant, whereas in the second case, we are taking the partial derivative with respect to keeping constant. In this case, when we increase slightly, the value of decreases to keep the total constant.
Here's the geometric interpretation:
 In the first case, where we are computing , we are geometrically computing the directional derivative along the positive direction, i.e., along a line with coordinate.
 In the second case, where we are computing , we are geometrically computing (up to scalar multiples) the directional derivative along lines with constant. These lines are downward sloping with a slope of .