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: 36


$$ f'(x) =(\cos^2x)^{\cos x}(-\sin x (\ln \cos^2x)+-2 \sin x).$$

Work Step by Step

Since $ f(x)=(\cos^2x)^{\cos x}$, applying $\ln $ on both sides, we get $$\ln f(x)=\ln (\cos^2x)^{\cos x}=\cos x \ln (\cos^2x) $$ Hence the derivative, using the product rule, is given by $$ f'/f =-\sin x (\ln \cos^2x)+\frac{\cos x}{\cos^2x}(\cos^2x)'\\ =-\sin x (\ln \cos^2x)+\frac{1}{\cos x} (-2\cos x\sin x)\\ =-\sin x (\ln \cos^2x)+-2\sin x $$ Then, we have $$ f'(x)=f(x)(-\sin x (\ln \cos^2x)+-2\sin x)\\ =(\cos^2x)^{\cos x}(-\sin x (\ln \cos^2x)+-2 \sin x).$$
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