Partial derivative

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:

Item	For partial derivative with respect to $x$	For partial derivative with respect to $y$
Notation	$\frac{\partial f (x, y)}{\partial x} \|_{(x, y) = (x_{0}, y_{0})}$ Also denoted $f_{x} (x_{0}, y_{0}$ or $f_{1} (x_{0}, y_{0})$	$\frac{\partial f (x, y)}{\partial y} \|_{(x, y) = (x_{0}, y_{0})}$ Also denoted $f_{y} (x_{0}, y_{0})$ or $f_{2} (x_{0}, y_{0})$
Definition as derivative	$\frac{d}{d x} f (x, y_{0}) \|_{x = x_{0}}$ . In other words, it is the derivative (at $x = x_{0}$ ) of the function $x \mapsto f (x, y_{0})$	$\frac{d}{d y} f (x_{0}, y) \|_{y = y_{0}}$ . In other words, it is the derivative (at $y = y_{0}$ ) of the function $y \mapsto f (x_{0}, y)$ .
Definition as limit (using derivative as limit of difference quotient)	$lim_{x \to x_{0}} \frac{f (x, y_{0}) - f (x_{0}, y_{0})}{x - x_{0}}$ $lim_{h \to 0} \frac{f (x_{0} + h, y_{0}) - f (x_{0}, y_{0})}{h}$	$lim_{y \to y_{0}} \frac{f (x_{0}, y) - f (x_{0}, y_{0})}{y - y_{0}}$ $lim_{h \to 0} \frac{f (x_{0}, y_{0} + h) - f (x_{0}, y_{0})}{h}$
Definition as directional derivative	Directional derivative at $(x_{0}, y_{0}$ with respect to a unit vector in the positive $x$ -direction.	Directional derivative at $(x_{0}, y_{0}$ with respect to a unit vector in the positive $y$ -direction.

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: