Partial derivative: Difference between revisions

Revision as of 00:29, 2 April 2012

Definition at a point

Generic definition

Suppose $f$ is a function of more than one variable, where $x$ is one of the input variables to $f$ . Fix a choice $x=x_{0}$ and fix the values of all the other variables. The partial derivative of $f$ with respect to $x$ , denoted $\partial f/\partial x$ , or $f_{x}$ , is defined as the derivative at $x_{0}$ of the function that sends $x$ to $f$ at $x$ for the same fixed choice of the other input variables.

For a function of two variables

Suppose $f$ is a real-valued function of two variables $x,y$ , i.e., the domain of $f$ is a subset of $\mathbb {R} ^{2}$ . We define the partial derivatives as follows:

Partial derivative with respect to $x$ :

${\frac {\partial f(x,y)}{\partial x}}|_{(x,y)=(x_{0},y_{0})}={\frac {d}{dx}}f(x,y_{0})|_{x=x_{0}}$

In words, it is the derivative at $x=x_{0}$ of the function $x\mapsto f(x,y_{0})$ .

This partial derivative is also denoted $f_{x}(x_{0},y_{0})$ or $f_{1}(x_{0},y_{0})$ .

Partial derivative with respect to $y$ :

${\frac {\partial f(x,y)}{\partial y}}|_{(x,y)=(x_{0},y_{0})}={\frac {d}{dy}}f(x_{0},y)|_{y=y_{0}}$

In words, it is the derivative at $y=y_{0}$ of the function $y\mapsto f(x_{0},y)$ .

This partial derivative is also denoted $f_{y}(x_{0},y_{0})$ or $f_{2}(x_{0},y_{0})$ .

For a function of multiple variables

The notation here gets a little messy, so read it carefully. We consider a function $f$ of $n$ variables, which we generically denote $(x_{1},x_{2},\dots ,x_{n})$ respectively. Consider a point $(a_{1},a_{2},\dots ,a_{n})$ in the domain of the function. In other words, this is a point where $x_{1}=a_{1},x_{2}=a_{2},\dots ,x_{n}=a_{n}$ .

Suppose $i$ is a natural number in the set $\{1,2,3,\dots ,n\}$ .

The partial derivative at this point $(a_{1},a_{2},\dots ,a_{n})$ with respect to the variable $x_{i}$ is defined as a derivative as follows:

${\frac {\partial }{\partial x_{i}}}f(x_{1},x_{2},\dots ,x_{n})|_{(x_{1},x_{2},\dots ,x_{n})=(a_{1},a_{2},\dots ,a_{n})}={\frac {d}{dx_{i}}}f(a_{1},a_{2},\dots ,a_{i-1},x_{i},a_{i+1},\dots ,a_{n})|_{x_{i}=a_{i}}$

In other words, it is the derivative (evaluated at $a_{i}$ ) of the function $x\mapsto f(x_{1},x_{2},\dots ,x_{i-1},a_{i},x_{i+1},\dots ,x_{n})$ with respect to $x_{i}$ , evaluated at the point $x_{i}=a_{i}$ .

@@ Line 31: / Line 31: @@
 Suppose <math>i</math> is a natural number in the set <math>\{ 1,2,3,\dots,n \}</math>.
-The '''partial derivative''' ''at'' this point <math>(a_1,a_2,\dots,a_n)</math> with respect to the variable <math>x_i</math> is defined as the [[derivative]]:
+The '''partial derivative''' ''at'' this point <math>(a_1,a_2,\dots,a_n)</math> with respect to the variable <math>x_i</math> is defined as a [[derivative]] as follows:
-<math>\frac{d}{dx_i}f(a_1,a_2,\dots,a_{i-1},x_i,a_{i+1}, \dots,a_n)|_{x_i = a_i}</math>
+<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)} = \frac{d}{dx_i}f(a_1,a_2,\dots,a_{i-1},x_i,a_{i+1}, \dots,a_n)|_{x_i = a_i}</math>
 In other words, it is the derivative (evaluated at <math>a_i</math>) of the function <math>x \mapsto f(x_1,x_2,\dots,x_{i-1},a_i,x_{i+1},\dots,x_n)</math> with respect to <math>x_i</math>, evaluated at the point <math>x_i = a_i</math>.