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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 $ x_i ,$ where the variables $ x_j$, $ 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 $ x_i$

$\displaystyle \partial_i f(x) = \lim_{h\to0}


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  automatically generated 8/ 4/2008