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: 28


$-\frac{1}{|\frac{x}{2}|\sqrt( {( {x^2})-4})}$

Work Step by Step

Given that $y$ = $cosec^{-1}{\frac{x}{2}}$ on substituting ${\frac{x}{2}}$ = $u$ 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 $\frac{dy}{dx}$ = $-\frac{1}{|(\frac{x}{2})|\sqrt ({({\frac{x}{2}})^2-1})}$ $\times$ $\frac{d {(\frac{x}{2})}}{dx}$ or $\frac{dy}{dx}$ = $-\frac{1}{|\frac{x}{2}|\sqrt( {(\frac{x^2}{4})-1})}$ $\times$ $\frac{1}{2}$ The final answer is: $-\frac{1}{|\frac{x}{2}|\sqrt( {( {x^2})-4})}$
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