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 $R^{2}$ . We define the partial derivatives as follows:

Partial derivative with respect to $x$ :

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

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 $y$ :

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

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 the derivative:

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

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