Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 2 - Derivatives - 2.6 Implicit Differentiation - 2.6 Exercises: 12

Answer

For $\cos (xy) =1+\sin y$ find $\frac{dy}{dx}$ by implicit differentiation $\frac{dy}{dx}=\frac{y \sin(xy)}{-x\sin (xy) -\cos y}$

Work Step by Step

Differentiate both sides with respect to $x$ using the chain rule and the trig rules $(-\sin (xy))(y+x\frac{dy}{dx})=(\cos(y))(\frac{dy}{dx})$ Isolate $\frac{dy}{dx}$ $-y\sin (xy)-x \sin(xy)\frac{dy}{dx}-\frac{dy}{dx}\cos y=0$ $\frac{dy}{dx}(-x\sin (xy) -\cos y)=y \sin(xy)$ $\frac{dy}{dx}=\frac{y \sin(xy)}{-x\sin (xy) -\cos y}$
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