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. They are useful in n-dimensional calculus and differential geometry.

The partial derivative of a function f with respect to the variable x is represented as or or f_x (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.

Another example involves the area A of a circle, though it only depends on the circle's radius r according to the formula

The partial derivative of A with respect to r is

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

Notation

For the following examples, let f be a function in x, y and z.

First-order partial derivatives:

Second-order partial derivatives:

Second-order mixed derivatives:

Higher-order partial and mixed derivatives:

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.