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 (as opposed to the total derivative, in which all variables are allowed to vary). 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 $\backslash frac\{\; \backslash partial\; f\; \}\{\; \backslash partial\; x\; \}$ or $\backslash partial\_xf$ or f_x (where $\backslash partial$ 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).

Examples

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

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

The partial derivative of V with respect to r is

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

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

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

and represents the rate with which the volume changes if its height is varied 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

$A\; =\; \backslash pi\; r^2\backslash ,$

The partial derivative of A with respect to r is

$\backslash frac\{\; \backslash partial\; A\}\{\backslash partial\; r\}\; =\; 2\; \backslash pi\; r.$

Equations involving an unknown function's partial derivatives are called partial differential equations and are common in physics and engineering.

Notation

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

First-order partial derivatives:

$\backslash frac\{\; \backslash partial\; f\}\{\; \backslash partial\; x\}\; =\; f\_x\; =\; \backslash partial\_x\; f$

Second-order partial derivatives:

$\backslash frac\{\; \backslash partial^2\; f\}\{\; \backslash partial\; x^2\}\; =\; f\_\{xx\}\; =\; \backslash partial\_\{xx\}\; f$

Second-order mixed derivatives:

$\backslash frac\{\; \backslash partial^2\; f\}\{\backslash partial\; x\backslash ,\backslash partial\; y\}\; =\; f\_\{xy\}\; =\; f\_\{yx\}\; =\; \backslash partial\_\{xy\}\; f\; =\; \backslash partial\_\{yx\}\; f$

Higher-order partial and mixed derivatives:

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

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

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

Formal definition and properties

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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 ith variable x_i as

$\backslash frac\{\; \backslash partial\; \}\{\backslash partial\; x\_i\; \}f(\backslash mathbf\{a\})\; =$

\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) exist 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 Clairaut's theorem:

$\backslash frac\{\backslash partial^2f\}\{\backslash partial\; x\_i\backslash ,\; \backslash partial\; x\_j\}\; =\; \backslash frac\{\backslash partial^2f\}\; \{\backslash partial\; x\_j\backslash ,\; \backslash partial\; x\_i\}.$

See also

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• directional derivative
• gradient
• curl
• divergence
• Jacobian
• exterior derivative
• Hessian matrix
• Laplacian
• d'Alembertian operator
• symmetry of second derivatives

Multivariate calculus

Derivada parcial | Partielle Ableitung | Derivada parcial | Dérivée partielle | 편미분 | Derivata parziale | נגזרת חלקית | Parciális derivált | Partiële afgeleide | 偏微分 | Pochodna cząstkowa | Derivada parcial | Частная производная | Osittaisderivaatta | Partiell derivata | Kısmi türev | 偏导数