Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 7: Transcendental Functions - Section 7.7 - Hyperbolic Functions - Exercises 7.7 - Page 430: 31

Answer

$- sech^{-1} x$

Work Step by Step

Given: $y=\cos^{-1} x-x sech^{-1}x$ Since, $\dfrac{d (\cos^{-1} x)}{dx}=-\dfrac{1}{\sqrt{1- x^2}}$ and $\dfrac{d (sech^{-1} x)}{dx}=\dfrac{-1}{x \sqrt{1- x^2}}$ Apply product rule to get the derivative: Thus, $\dfrac{dy}{dx}=[-\dfrac{1}{\sqrt{1- x^2}}] [ sech^{-1} x (1) +(x) \dfrac{-1}{x \sqrt{1- x^2}}]$ or, $=-\dfrac{1}{\sqrt{1- x^2}}- sech^{-1} x+ \dfrac{1}{ \sqrt{1- x^2}}$ Hence, $\dfrac{dy}{dx}=- sech^{-1} x$
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