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: 27

Answer

$$ y' = \frac{6x ( \sinh^{-1} ( x^2))^2}{\sqrt{1+(x^2)^2}}.$$

Work Step by Step

Since $ y=( \sinh^{-1} ( x^2))^3$, then the derivative, by using the chain rule, is given by $$ y'= 3 ( \sinh^{-1} ( x^2))^2( \sinh^{-1} ( x^2))'=3 ( \sinh^{-1} ( x^2))^2 \frac{(x^2)'}{\sqrt{1+(x^2)^2}}\\ = \frac{6x ( \sinh^{-1} ( x^2))^2}{\sqrt{1+(x^2)^2}}.$$
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