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: 30


$$ y'= \frac{x }{\sqrt{1+x^2}\sqrt{2+x^2}}.$$

Work Step by Step

Since $ y=\sinh^{-1} (\sqrt{x^2+1})$, then the derivative, by using the chain rule and the fact that $\frac{d}{d x} \sinh ^{-1} u=\frac{u'}{\sqrt{u^{2}+1}}$, is given by $$ y'= \frac{1}{\sqrt{1+(\sqrt{x^2+1})^2}}(\sqrt{1+x^2})'=\frac{2x }{2\sqrt{1+x^2}\sqrt{2+x^2}}\\ =\frac{x }{\sqrt{1+x^2}\sqrt{2+x^2}}.$$
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