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 , … ) {\displaystyle f(x,y,\dots )} with respect to the variable x {\displaystyle x} (analogously for any other variable) is variously denoted by
It is the rate of change of the function in the x {\displaystyle x} -direction. Sometimes, for z = f ( x , y , … ) {\displaystyle z=f(x,y,\ldots )} , the partial derivative of z {\displaystyle z} with respect to x {\displaystyle x} is denoted as ∂ z ∂ x . {\displaystyle {\tfrac {\partial z}{\partial x}}.} Since a partial derivative generally has the same arguments as the original function, its functional dependence is sometimes explicitly signified by the notation, such as in:
f x ′ ( x , y , … ) , ∂ f ∂ x ( x , y , … ) . {\displaystyle f'_{x}(x,y,\ldots ),{\frac {\partial f}{\partial x}}(x,y,\ldots ).}
The symbol used to denote partial derivatives is ∂. One of the first known uses of this symbol in mathematics is by Marquis de Condorcet from 1770, who used it for partial differences. The modern partial derivative notation was created by Adrien-Marie Legendre (1786), although he later abandoned it; Carl Gustav Jacob Jacobi reintroduced the symbol in 1841.
Definition Like ordinary derivatives, the partial derivative is defined as a limit. Let U be an open subset of R n {\displaystyle \mathbb {R} ^{n}} and f : U → R {\displaystyle f:U\to \mathbb {R} } a function. The partial derivative of f at the point a = ( a 1 , … , a n ) ∈ U {\displaystyle \mathbf {a} =(a_{1},\ldots ,a_{n})\in U} with respect to the i-th variable xi is defined as
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