Mathematics-Online lexicon: Partial Derivative

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	Mathematics-Online lexicon:
	Partial Derivative

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The partial derivative $\partial_i f$ of a function $f = f(x_1, \ldots , x_n)$ is the derivative of the function

$\displaystyle x_i \mapsto f(x_1,\ldots,x_i,\ldots,x_n)\, ,$

with respect to

where the variables

, $j\ne i$ are regarded as constants.

Other notations are

$\displaystyle \partial_i f = f_{x_i} = \frac{\partial f}{\partial x_i}\, .$

Note that by definition of the derivative with respect to

$\displaystyle \partial_i f(x) = \lim_{h\to0} \frac{f(\ldots,x_i+h,\ldots)-f(\ldots,x_i,\ldots)}{h} \,.$

()

Examples:

automatically generated 8/ 4/2008