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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. 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 fx (where <math>\partial<math> is a rounded 'd' known as the 'partial derivative symbol,' which coincides with the cursive Cyrillic letter "de" and is pronounced as its English couterpart "d" - that incidentally was the notation first introduced by Legendre).
If f is a function of x1, ..., xn and dx1, ..., dxn are thought of as infinitely small increments of x1, ..., xn 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>
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
- <math>\left( \frac{\partial f}{\partial x} \right)_{y,z}<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,...,an)∈U with respect to the i-th variable xi 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/∂xi(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/∂xi 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
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