University Calculus: Early Transcendentals (3rd Edition)

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

Chapter 3 - Section 3.7 - Implicit Differentiation - Exercises - Page 164: 11



Work Step by Step

$$x+\tan(xy)=0$$ 1) Differentiate both sides of the equation with respect to $x$: $$\frac{dx}{dx}+\frac{d}{dx}(\tan(xy))=0$$ $$1+\sec^2(xy)\frac{d}{dx}(xy)=0$$ $$1+\sec^2(xy)(y+x\frac{dy}{dx})=0$$ $$1+y\sec^2(xy)+x\sec^2(xy)\frac{dy}{dx}=0$$ 2) Collect all the terms with $dy/dx$ onto one side and solve for $dy/dx$: $$x\sec^2(xy)\frac{dy}{dx}=-(y\sec^2(xy)+1)$$ $$\frac{dy}{dx}=-\frac{y\sec^2(xy)+1}{x\sec^2(xy)}$$
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