Calculus (3rd Edition)

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

Chapter 8 - Techniques of Integration - 8.4 Integrals Involving Hyperbolic and Inverse Hyperbolic Functions - Exercises - Page 415: 45


$$\frac{d}{d y} g d(y) =\operatorname{sech} y$$

Work Step by Step

Since $$g d(y)=\tan ^{-1}(\sinh y)$$ Then \begin{aligned} \frac{d}{d y} g d(y) &=\frac{d}{d y} \tan ^{-1}(\sinh y) \\ &=\frac{1}{1+\sinh ^{2} y} \cosh y \\ &=\frac{1}{\cosh ^{2} y} \cosh y \\ &=\frac{1}{\cosh y}\\ &=\operatorname{sech} y \end{aligned}
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