Partial differentiation

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 ${\displaystyle f(x,y,\dots )}$ with respect to the variable ${\displaystyle x}$ is variously denoted by

Sometimes, if ${\displaystyle z=f(x,y,\dots ),}$ the partial derivative of ${\displaystyle z}$ with respect to ${\displaystyle x}$ is denoted as as ${\displaystyle {\partial z} \over {\partial x}.}$ Since a partial derivative is generally a function of the same arguments as was the original function, this functional dependence is sometimes explicitly included in the notation, as in

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