Partial derivative

Where a function depends on more than one variable, the concept of a partial derivative is used. Partial derivatives can be thought of informally as taking the derivative of the function with all but one variable held temporarily constant near a point.

The partial derivative of a function f with respect to the variable x is represented as f_x or <math>\frac{ \partial f }{ \partial x }</math> (where <math>\partial</math> is a rounded 'd' known as the 'partial derivative symbol').

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

<math>V = \frac{ r^2 h \pi }{3}</math>

The partial derivative of V with respect to r is

<math>\frac{ \partial V}{\partial r} = \frac{ 2r h \pi }{3}</math>

it describes the rate with which a cone's volume changes if its radius is increased and its height is kept constant. The partial with respect to h is

<math>\frac{ \partial V}{\partial h} = \frac{ r^2 \pi }{3}</math>

and represents the rate with which the volume changes if its height is increased and its radius is kept constant.

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 Rⁿ and f : U -> R a function. We define the partial derivative of f at the point a=(a₁,...,a_n)∈U with respect to the i-th variable x_i as

<math>\frac{ \partial }{\partial x_i }f(a) =

\lim_{h \rightarrow 0}{ f(a_1, ... , a_{i-1}, a_i+h, a_{i+1},...,a_n) - f(a_1,...,a_n) \over h } </math>

Even if all partial derivatives ∂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 C¹ 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 C² function; in this case, the partial derivatives can be exchanged:

<math>\frac{\partial^2f}{\partial x_i} \partial x_j = \frac{\partial^2f} {\partial x_j} \partial x_i</math>

The vector consisting of all partial derivatives of f at a given point a is called the gradient of f at a:

<math>\operatorname{grad}f(a) = \left( \frac{\partial f}{\partial x_1}(a), \dots , \frac{\partial f}{\partial x_n}(a) \right)</math>

If f is a C¹ function, then grad f(a) has a geometrical interpretation: it is the direction in which f grows the fastest, the direction of steepest ascent.