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: 15



Work Step by Step

$$e^{2x}=\sin(x+3y)$$ 1) Differentiate both sides of the equation with respect to $x$: $$\frac{d}{dx}(e^{2x})=\frac{d}{dx}\sin(x+3y)$$ $$e^{2x}\frac{d}{dx}(2x)=\cos(x+3y)\frac{d}{dx}(x+3y)$$ $$2e^{2x}=\cos(x+3y)(1+3\frac{dy}{dx})$$ $$2e^{2x}=\cos(x+3y)+3\cos(x+3y)\frac{dy}{dx}$$ 2) Collect all the terms with $dy/dx$ onto one side and solve for $dy/dx$: $$3\cos(x+3y)\frac{dy}{dx}=2e^{2x}-\cos(x+3y)$$ $$\frac{dy}{dx}=\frac{2e^{2x}-\cos(x+3y)}{3\cos(x+3y)}$$
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