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Elementary Calculus Partial Differentiation Partial Derivates Definition
 

Definition

Here is a precise definition.

DEFINITION

The partial derivatives of f(x, y) at the point (a, b) are the limits

11_partial_differentiation-158.gif

 

11_partial_differentiation-161.gif

A partial derivative is undefined if the limit does not exist.

When fx(a, b) exists, it is equal to the standard part

11_partial_differentiation-162.gif

for any nonzero infinitesimal Dx. Similarly when fy(a, b) exists,

11_partial_differentiation-163.gif

for any nonzero infinitesimal Dy.

Just as the one-variable derivative f'(x) is a function of x, the partial derivatives fx(x, y) and fy(x, y) are again functions of x and y. At each point (x, y), the partial derivative fx(x, y) either has exactly one value or is undefined.

Last Update: 2006-Nov-05