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

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

is standard,^[1][2][3] but other notations are sometimes used.

Examples

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

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

Or, we can do the following:

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

Related pages

• Difference quotient
• Partial differential equation