# Difference between revisions of "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 $x$:

$\frac{\partial f(x,y)}{\partial x}|_{(x,y) = (x_0,y_0)} = \frac{d}{dx}f(x,y_0)|_{x = x_0}$

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 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): (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:

$\frac{d}{dx_i}f(a_1,a_2,\dots,a_{i-1},x_i,a_{i+1}, \dots,a_n)|_{x_i = a_i}$

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 $x_i = a_i$.