Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 6 - Inverse Functions - 6.4* General Logarithmic and Exponential Functions - 6.4* Exercise - Page 464: 44

Answer

$$f''(x)=x^{\cos(x)} \left[\frac{\cos(x)}{x}-\sin(x)\ln (x)\right]$$

Work Step by Step

Given $$f(x)= x^{\cos(x)}$$ Take $\ln $ for both sides \begin{aligned} \ln f(x)&= \ln x^{\cos (x)}\\ &=\cos (x)\ln (x) \end{aligned} Differentiate both sides \begin{aligned} \frac{f'(x)}{f(x)}&= \cos(x) \frac{1}{x} + \ln (x) (-\sin(x))\\ f'(x)&= f(x) \left[\frac{\cos(x)}{x}-\sin(x)\ln (x)\right]\\ &=x^{\cos(x)} \left[\frac{\cos(x)}{x}-\sin(x)\ln (x)\right] \end{aligned} we can note that $f(x)$ is increasing when $f'(x)>0 $ and $f(x)$ is decreasing when $f'(x)<0 $
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