Partial derivative

In multivariable calculus, the partial derivative of a function is the derivative of one variable when all other variables are held constant. In other words, a partial derivative takes the derivative of certain variables of a function while not differentiating other variable(s). Partial derivatives are often used in multivariable functions.

For partial derivatives of function f with respect to variable x, the notation

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

is standard, but other notations are sometimes used.

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]

Partial Derivative Media

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  The volume of a cone depends on height and radius

Related pages

• Difference quotient
• Partial differential equation