Partial derivative

In calculus (particularly in multivariate calculus, the study of rate of change on functions with multiple variables), the partial derivative of a function is the derivative of one named variable, where all other unnamed variables of the function are held constant. In other words, the partial derivative takes the derivative of certain indicated variables of a function, and does not differentiate the other variable(s).

For the partial derivative of a function f with respect to the variable x, the notations

[math]\displaystyle{ \frac{\partial f}{\partial x} }[/math], [math]\displaystyle{ f_x }[/math], [math]\displaystyle{ \partial_x f }[/math]

are usually used, although other notations are valid. Usually, although not always, the partial derivative is taken in a multivariable function (a function which takes two or more variables as input).

Examples

If we have a function [math]\displaystyle{ f(x, y)=x^2+y }[/math], then there are several partial derivatives of f(x, y) that are all equally valid. For example,

[math]\displaystyle{ \frac{\partial }{\partial y}[f(x, y)]=1 }[/math]

Or, we can do the following:

[math]\displaystyle{ \frac{\partial}{\partial x}[f(x, y)]=2x }[/math]

Related pages

• Difference quotient
• Partial differential equation