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 154: 60



Work Step by Step

Implicitly differentiating $x^{3}+3xy^{2}=1$, we have $3x^{2}\times\frac{dx}{dt}+3(\frac{dx}{dt}\times y^{2}+x\times2y\frac{dy}{dt})=0$ $\implies 3x^{2}\frac{dx}{dt}+3y^{2}\frac{dx}{dt}+6xy\frac{dy}{dt}=0$ $\implies 6xy\frac{dy}{dt}=-3x^{2}\frac{dx}{dt}-3y^{2}\frac{dx}{dt}$ $\implies 6xy\frac{dy}{dt}=(-3x^{2}-3y^{2})\frac{dx}{dt}$ $\implies \frac{dy}{dt}=(\frac{-3x^{2}-3y^{2}}{6xy})\frac{dx}{dt}$ $=(\frac{-x^{2}-y^{2}}{2xy})\frac{dx}{dt}$
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