Partial derivative

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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). Partial derivatives are useful in vector calculus and differential geometry.

The partial derivative of a function f with respect to the variable x is written as f_x or ∂f/∂x. The partial-derivative symbol ∂ is a rounded letter, distinguished from the straight d of total-derivative notation. The notation was introduced by Legendre and gained general acceptance after its reintroduction by Jacobi.

1 Examples
2 Notation
3 Formal definition and properties
4 See also

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

$V(r, h) = \frac{\pi r^2 h}{3}.$

The partial derivative of V with respect to r is

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

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

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

and represents the rate with which the volume changes if its height is varied and its radius is kept constant.

Now consider by contrast the total derivative of V with respect to r and h. They are, respectively

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

and

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

We see that the difference between the total and partial derivative is the elimination of indirect dependencies between variables in the latter.

Equations involving an unknown function's partial derivatives are called partial differential equations and are common in physics, engineering, and other sciences and applied disciplines.

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:

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

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

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

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) 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. We can use this fact to generalize for vector valued functions (f : U → R'^m) by carefully using a componentwise argument.

The partial derivative $\frac{\partial f}{\partial x}$ can be seen as another function defined on U and can again be partially differentiated. If all mixed second order partial derivatives are continuous at a point (or on a set), we call f a C² function at that point (or on that set); in this case, the partial derivatives can be exchanged by Clairaut's theorem: