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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 vector calculus and differential geometry.

The partial derivative of a function f with respect to the variable x is represented as $\frac{ \partial f }{ \partial x }$ or $\partial_xf$ or f_x (where $\partial$ is a rounded 'd' known as the 'partial derivative symbol,' which coincides with the cursive Cyrillic letter "de" and is pronounced as its English counterpart "d" - that incidentally was the notation first introduced by Legendre).

Contents

1 Examples

2 Notation

3 Formal definition and properties

4 See also

Examples

Consider the volume V of a cone; it depends on the cone's height h and its radius r according to the formula

$V = \frac{ r^2 h \pi }{3}$

The partial derivative of V with respect to r is

$\frac{ \partial V}{\partial r} = \frac{ 2r h \pi }{3}$

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

$\frac{ \partial V}{\partial h} = \frac{ r^2 \pi }{3}$

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

A = π r 2

The partial derivative of A with respect to r is

$\frac{ \partial A}{\partial r} = 2 \pi r$

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:

$\frac{ \partial f}{ \partial x} = f_x = \partial_x f$

Second-order partial derivatives:

$\frac{ \partial^2 f}{ \partial x^2} = f_{xx} = \partial_{xx} f$

Second-order mixed derivatives:

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

Higher-order partial and mixed derivatives:

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

When dealing with functions of multiple variables, some of these variables may be related to each other, and it may be necessary to specify explicitly which variables are being held constant. In fields such as statistical mechanics, the partial derivative of f with respect to x, holding y and z constant, is often expressed as

$\left( \frac{\partial f}{\partial x} \right)_{y,z}$

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

$\frac{ \partial }{\partial x_i }f(\mathbf{a}) = \lim_{h \rightarrow 0}{ f(a_1, \dots , a_{i-1}, a_i+h, a_{i+1}, \dots ,a_n) - f(a_1, \dots ,a_n) \over h }$

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 by the equality of mixed partials theorem:

$\frac{\partial^2f}{\partial x_i\, \partial x_j} = \frac{\partial^2f} {\partial x_j\, \partial x_i}.$

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