Calculus (3rd Edition)

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

Chapter 7 - Exponential Functions - 7.9 Hyperbolic Functions - Exercises - Page 384: 16


$$ y'= 5\tanh (x ) (\ln \cosh(x))^4$$

Work Step by Step

Recall that $(\ln x)'=\dfrac{1}{x}$ Recall that $(\cosh x)'=\sinh x$ Since $ y=(\ln \cosh(x))^5$, then the derivative, by using the chain rule, is given by $$ y'=5(\ln \cosh(x))^4(\ln \cosh(x))'=5(\ln \cosh(x))^4(\frac{1}{\cosh x})(\cosh x)'\\ =5(\ln \cosh(x))^4\frac{\sinh x}{\cosh x}=5\tanh (x ) (\ln \cosh(x))^4.$$
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