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

${\frac {\partial f}{\partial x}}$, $f_{x}$, $\partial _{x}f$

is standard, but other notations are sometimes used.

Examples

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

${\frac {\partial }{\partial y}}[f(x,y)]=1$

Or, we can do the following:

${\frac {\partial }{\partial x}}[f(x,y)]=2x$

Related pages

• Difference quotient
• Partial differential equation