Partial derivative

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This editable Main Article is under development and subject to a disclaimer.

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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[edit]

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

∂ f ∂ x i ( t 1 , … , t n ) = lim h -> 0 f ( t 1 , … , t i + h , … , t n ) − f ( t 1 , … , t n ) h {\displaystyle {\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[edit]

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

## See also[edit]

* Derivative
* Total derivative