Partial derivative

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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

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{\part f}{\part 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 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

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