Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 3 - Differentiation - 3.8 Implicit Differentiation - Exercises - Page 152: 30



Work Step by Step

Given $$\sin ^{2}(3 y)=x+y, \quad\left(\frac{2-\pi}{4}, \frac{\pi}{4}\right)$$ Differentiate with respect to $x$ \begin{align*} 2\sin(3 y)\cos(3y)y'&=1+y'\\ ( \sin(6y)-1)y'&= 1 \end{align*} Then \begin{aligned} y'&= \frac{1}{\sin(6y)-1}\\ \left.y^{\prime}\right|_{\left(\dfrac{2-\pi}{4}, \dfrac{\pi}{4}\right)} &=\frac{1}{\sin(3\pi/2)-1} \\ &= \frac{-1}{4} \end{aligned}
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