Calculus (3rd Edition)

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

Chapter 7 - Exponential Functions - 7.3 Logarithms and Their Derivatives - Exercises - Page 343: 74


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

Work Step by Step

Recall that $(e^x)'=e^x$ Recall that $(\ln x)'=\dfrac{1}{x}$ Recall that $(\sin x)'=\cos x$. Recall the product rule: $(uv)'=u'v+uv'$ Using the method from Example 9, we get: $$ f(x)=(e^{\ln x})^{\cos x}=e^{\cos x\ln x}.$$ Now taking the derivative, we get $$ f'(x)= e^{\cos x\ln x}(\cos x\ln x)'=e^{\cos x\ln x}(-\sin x \ln x+\cos x /x) =e^{\cos x\ln x}(-\sin x \ln x+\cos x/x)=x^{\cos x}(-\sin x\ln x+\frac{\cos x}{x}).$$
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