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: 51


$$\frac{d}{d t} \sinh ^{-1} t=\frac{1}{\sqrt{t^{2}+1}} $$

Work Step by Step

Let \begin{align*} x&=\sinh ^{-1} t \Rightarrow t=\sinh x \end{align*} then \begin{align*} \frac{d t}{d x}&=\cosh x\\ \frac{d x}{d t}&=\frac{1}{\cosh x}\\ &=\frac{1}{\sqrt{\sinh^2x+1}}\\ &=\frac{1}{\sqrt{t^2 +1}} \end{align*} Hence, verified.
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