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Power Rule General Form

The Inverse Function Rule can be proved from the Chain Rule as follows.

Let

y = f(x), x = g(y)

be inverse functions whose derivatives exist. Then

02_differentiation-353.gif

whence

02_differentiation-354.gif

Using the Chain Rule we may write the Power Rule in a general form.

POWER RULE

Let r be a rational number, and let u depend on x. If u > 0 and du/dx exists, then

02_differentiation-355.gif

This is proved by letting y = ur and computing dy/dx by the Chain Rule,

02_differentiation-356.gif
Home Differentiation Chain Rule Power Rule General Form

Last Update: 2006-11-05