Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 7 - Exponential Functions - Chapter Review Exercises - Page 386: 37


$$ G'(t)=(\sin^2t)^t(\ln (\sin^2t)+2t\cot t) .$$

Work Step by Step

Since $ G(t)=(\sin^2t)^t $, applying $\ln $ on both sides, we get $$\ln G(t)=\ln (\sin^2t)^t=t \ln (\sin^2t)$$ Hence the derivative, using the product rule, is given by $$ G'/G = \ln (\sin^2t)+\frac{t}{\sin^2t}(\sin^2t)'= \ln (\sin^2t)+\frac{t}{\sin^2t}(2\sin t \cos t) \\ =\ln (\sin^2t)+2t\cot t.$$ Then, we have $$ G'(t)=G(t)(\ln (\sin^2t)+2t\cot t)=(\sin^2t)^t(\ln (\sin^2t)+2t\cot t) .$$
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