In mathematics, a

**partial derivative**of a function of several variables is its derivative with respect to one of those variables with the others held constant.

The partial derivative of a function *f* with respect to the variable *x* is represented as *f\*_{x} or (where is a rounded 'd' known as the 'partial derivative symbol').

If *f* is a function of *x*_{1}, ..., *x*_{n} and *dx*_{1}, ..., *dx*_{n} are thought of as infinitely small increments of *x*_{1}, ..., *x*_{n} respectively, then the corresponding infinitely small increment of *f* is

*f*; each term in the sum is a "partial differential" of

*f*.

As an example, consider the volume *V* of a cone; it depends on the cone's height *h* and its radius *r* according to the formula

*V*with respect to

*r*is

*h*is

Equations involving an unknown function's partial derivatives are called partial differential equations and are ubiquitous throughout science.

## Formal definition and properties

Like ordinary derivatives, the partial derivative is defined as a limit. Let*U*be an open subset of

**R**

^{n}and

*f*:

*U*

`->`

**R**a function. We define the partial derivative of

*f*at the point

*a*=(

*a*

_{1},...,

*a*

_{n})∈

*U*with respect to the

*i*-th variable

*x*

_{i}as

*f*/∂

*x*

_{i}(

*a*) exists at a given point

*a*, the function need not be continuous there. However, if all partial derivatives exist in a neighborhood of

*a*and are continuous there, then

*f*is totally differentiable in that neighborhood and the total derivative is continuous. In this case, we say that

*f*is a C

^{1}function.

The partial derivative ∂*f*/∂*x*_{i} can be seen as another function defined on *U* and can again be partially differentiated. If all mixed partial derivatives exist and are continuous, we call *f* a C^{2} function; in this case, the partial derivatives can be exchanged:

*f*at a given point

*a*is called the gradient of

*f*at

*a*:

*f*is a C

^{1}function, then grad

*f*(

*a*) has a geometrical interpretation: it is the direction in which

*f*grows the fastest, the direction of

*steepest ascent*.

See also: Directional derivative