University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 3 - Section 3.9 - Inverse Trigonometric Functions - Exercises - Page 181: 27

Answer

$-\frac{2}{|x^2+1|\sqrt( {( {x^2})+2})}$

Work Step by Step

Given that $y$ = $cosec^{-1}(x^2+1)$ where $x\gt0$ on substituting ${x^2+1}$ = $u$ on applying chain rule derivative $\frac{dy}{dx}$ = $\frac{{dcosec^{-1}{u}}}{du}$ $\times$ $\frac{du}{dx}$ or $\frac{dy}{dx}$ = $-\frac{1}{{|u|}\sqrt ({u^2}-1)}$ $\times$ $\frac{du}{dx}$ so $\frac{dy}{dx}$ = $-\frac{1}{|(x^2+1)|\sqrt ({({x^2+1})^2-1})}$ $\times$ $\frac{d {(x^2+1)}}{dx}$ or $\frac{dy}{dx}$ = $-\frac{1}{|x^2+1|\sqrt( {({x^4})+2x^2})}$ $\times$ $2x$ The final answer is: $-\frac{2}{|x^2+1|\sqrt( {( {x^2})+2})}$
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