Partial derivative: Difference between revisions

Revision as of 00:38, 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 $R^{2}$ . Suppose $(x_{0}, y_{0})$ is a point in the domain of $f$ . We define the partial derivatives at $(x_{0}, y_{0})$ as follows:

Which partial derivative?	Notation for partial derivative	Definition as derivative	Definition as limit
Partial derivative with respect to $x$	$\frac{\partial f (x, y)}{\partial x} \|_{(x, y) = (x_{0}, y_{0})}$ Also denoted $f_{x} (x_{0}, y_{0}$ or $f_{1} (x_{0}, y_{0})$	$\frac{d}{d x} f (x, y_{0}) \|_{x = x_{0}}$ . In other words, it is the derivative (at $x = x_{0}$ ) of the function $x \mapsto f (x, y_{0})$	$lim_{x \to x_{0}} \frac{f (x, y_{0}) - f (x_{0}, y_{0})}{x - x_{0}}$ $lim_{h \to 0} \frac{f (x_{0} + h, y_{0}) - f (x_{0}, y_{0})}{h}$
Partial derivative with respect to $y$	$\frac{\partial f (x, y)}{\partial y} \|_{(x, y) = (x_{0}, y_{0})}$ Also denoted $f_{y} (x_{0}, y_{0})$ or $f_{2} (x_{0}, y_{0})$	$\frac{d}{d y} f (x_{0}, y) \|_{y = y_{0}}$ . In other words, it is the derivative (at $y = y_{0}$ ) of the function $y \mapsto f (x_{0}, y)$ .	$lim_{y \to y_{0}} \frac{f (x_{0}, y) - f (x_{0}, y_{0})}{y - y_{0}}$ $lim_{h \to 0} \frac{f (x_{0}, y_{0} + h) - f (x_{0}, y_{0})}{h}$

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 given below.

This partial derivative is also denoted as $f_{x_{i}} (a_{1}, a_{2}, \dots, a_{n})$ or $f_{i} (a_{1}, a_{2}, \dots, a_{n})$ .

As a derivative:

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

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 $x_{i} = a_{i}$ .

As a limit: The partial derivative can be defined explicitly as a limit:

@@ Line 7: / Line 7: @@
 ===For a function of two variables===
-Suppose <math>f</math> is a real-valued 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 real-valued 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>. We define the partial derivatives at <math>(x_0,y_0)</math> as follows:
-* '''Partial derivative with respect to <math>x</math>''':
+{| class="sortable" border="1"
+! Which partial derivative? !! Notation for partial derivative !! Definition as derivative !! Definition as limit
-<math>\frac{\partial f(x,y)}{\partial x}|_{(x,y) = (x_0,y_0)} = \frac{d}{dx}f(x,y_0)|_{x = x_0}</math>
+|-
+| Partial derivative with respect to <math>x</math> || <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{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>\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>
-In words, it is the [[derivative]] at <math>x = x_0</math> of the function <math>x \mapsto f(x,y_0)</math>.
+|-
+| Partial derivative with respect to <math>y</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> ||  <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>. || <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>
-This partial derivative is also denoted <math>f_x(x_0,y_0)</math> or <math>f_1(x_0,y_0)</math>.
+|}
-* '''Partial derivative with respect to <math>y</math>''':
-<math>\frac{\partial f(x,y)}{\partial y}|_{(x,y) = (x_0,y_0)} = \frac{d}{dy}f(x_0,y)|_{y = y_0}</math>
-In words, it is the [[derivative]] at <math>y = y_0</math> of the function <math>y \mapsto f(x_0,y)</math>.
-This partial derivative is also denoted <math>f_y(x_0,y_0)</math> or <math>f_2(x_0,y_0)</math>.
 ===For a function of multiple variables===
@@ Line 31: / Line 23: @@
 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 a [[derivative]] as follows:
+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 given below.
+This partial derivative is also denoted as <math>f_{x_i}(a_1,a_2,\dots,a_n)</matH> or <math>f_i(a_1,a_2,\dots,a_n)</math>.
+'''As a derivative''':
 <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>.
+'''As a limit''': The partial derivative can be defined explicitly as a limit: