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Partial derivation

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 <math>\frac{ \partial f }{ \partial x }<math> or <math>\partial_xf<math> or f_x (where <math>\partial<math> 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

<math>df=\frac{\partial f}{\partial x_1}\,dx_1+\cdots+\frac{\partial f}{\partial x_n}\,dx_n.<math>

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

<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.

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

<math>A = \pi r^2<math>

The partial derivative of A with respect to r is

<math>\frac{ \partial A}{\partial r} = 2 \pi r<math>

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:

<math>\frac{ \partial f}{ \partial x} = f_x = \partial_x f<math>

Second-order partial derivatives:

<math>\frac{ \partial^2 f}{ \partial x^2} = f_{xx} = \partial_{xx} f<math>

Second-order mixed derivatives:

<math>\frac{ \partial^2 f}{\partial x\,\partial y} = f_{xy} = f_{yx} = \partial_{xy} f = \partial_{yx} f<math>

Higher-order partial and mixed derivatives:

<math>\frac{ \partial^{i+j+k} f}{ \partial x^i\, \partial y^j\, \partial z^k } = f^{(i, j, k)}<math>

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(\mathbf{a}) =

\lim_{h \rightarrow 0}{ f(a_1, \dots , a_{i-1}, a_i+h, a_{i+1}, \dots ,a_n) - f(a_1, \dots ,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.

See also:

• directional derivative
• gradient
• symmetry of second derivatives.