Partial derivative

In multivariable calculus, the partial derivative of a function is the derivative of one variable in the function while supposing all other variables to act as constants whose derivative is zero. In other words, a partial derivative essentially holds all variables constant except the one being differentiated. 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,

${\frac {\partial }{\partial y}}[f(x,y)]=1$ {In the partial derivative of y, x is ignored/considered a constant whose derivative is zero}

Or, we can do the following:

${\frac {\partial }{\partial x}}[f(x,y)]=2x$ {In the partial derivative of x, y is ignored/considered a constant whose derivative is zero}

Related pages

• Difference quotient
• Partial differential equation