Partial derivative

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 x₁, ..., x_n and dx₁, ..., dx_n are thought of as infinitely small increments of x₁, ..., x_n respectively, then the corresponding infinitely small increment of f is

That quantity is the "total differential" of 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

The partial derivative of V with respect to r is

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

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

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:

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

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.