Partial derivative

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables while all others are kept constant. Partial derivatives are widely used in differential geometry, vector calculus, and physics.

Definition

A function $f(x_{1},\dots ,x_{n})$ is called a function of multiple variables if $n>1$. The partial derivative of $f$ in the direction $x_{i}$ at the point $(t_{1},\dots ,t_{n})$ is defined as

${\frac {\partial f}{\partial x_{i}}}(t_{1},\dots ,t_{n})=\lim _{h\rightarrow 0}{\frac {f(t_{1},\dots ,t_{i}+h,\dots ,t_{n})-f(t_{1},\dots ,t_{n})}{h}}$

Notation

The partial derivative of a function f with respect to the variable x_i is written as f_xi or ∂f/∂x_i. The partial derivative symbol ∂ is distinguished from the straight d that denotes the total derivative.

See also

• Derivative
• Total derivative