What is "partial derivative"

n. (context mathematics English) A derivative with respect to one variable of a function of several variables.

n. the derivative of a function of two or more variables with respect to a single variable while the other variables are considered to be constant [syn: partial]

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 used in vector calculus and differential geometry.

The partial derivative of a function f(x, y, ...) with respect to the variable x is variously denoted by

$f^\prime_x,\ f_x,\ \partial_x f, \frac{\partial}{\partial x}f, \text{ or } \frac{\partial f}{\partial x}.$

Since in general a partial derivative is a function of the same arguments as was the original function, this functional dependence is sometimes explicitly included in the notation, as in

$$f_x(x, y, ...), \ \frac{{\partial f}}{{\partial x}} (x, y, ...).$$

The partial-derivative symbol is ∂. One of the first known uses of the symbol in mathematics is by Marquis de Condorcet from 1770, who used it for partial differences. The modern partial derivative notation is by Adrien-Marie Legendre (1786), though he later abandoned it; Carl Gustav Jacob Jacobi re-introduced the symbol in 1841.