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: 28


$$y'=-\frac{4(csch^{-1} ( 3x))^3}{|x|\sqrt{(3x)^2-1}}.$$

Work Step by Step

Recall that $(\csc^{-1} x)'=\dfrac{-1}{|x|\sqrt{x^2-1}}$ Since $ y=( csch^{-1} ( 3x))^4$, then the derivative, by using the chain rule, is given by $$ y'= 4 (csch^{-1} ( 3x))^3( csch^{-1} (3 x))'=4 (csch^{-1} ( 3x))^3\frac{-3}{|3x|\sqrt{(3x)^2-1}}\\ =-\frac{12(csch^{-1} ( 3x))^3}{|3x|\sqrt{(3x)^2-1}}=-\frac{4(csch^{-1} ( 3x))^3}{|x|\sqrt{(3x)^2-1}}.$$
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