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Home Exponential and Logartihmic Functions Derivatives and Integrals Involving ln x Theorem 2 (Power Rule)
 
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Theorem 2 (Power Rule)

The Power Rule d(xr) = r xr-1 dx was proved in Chapter 2 only when the exponent r is rational. We can use logarithmic differentiation to show that the Power Rule holds even for irrational r.

THEOREM 2

(Power Rule)

Let b be any real number. Then

08_exp-log_functions-373.gif

PROOF

Let y = xb. Then In y = b ln x, and

08_exp-log_functions-374.gif

The formula ∫ 1/x dx = ln |x| + C allows us to integrate a number of basic functions which we could not handle before.
Home Exponential and Logartihmic Functions Derivatives and Integrals Involving ln x Theorem 2 (Power Rule)

Last Update: 2006-11-16