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: 18



Work Step by Step

$$r-2\sqrt\theta=\frac{3}{2}\theta^{2/3}+\frac{4}{3}\theta^{3/4}$$ $$r=2\sqrt\theta+\frac{3}{2}\theta^{2/3}+\frac{4}{3}\theta^{3/4}$$ $$r=2\theta^{1/2}+\frac{3}{2}\theta^{2/3}+\frac{4}{3}\theta^{3/4}$$ We do not have to use implicit differentiation here, but apply the derivative rules right away: $$\frac{dr}{d\theta}=2\times\frac{1}{2}\theta^{-1/2}+\frac{3}{2}\times\frac{2}{3}\theta^{-1/3}+\frac{4}{3}\times\frac{3}{4}\theta^{-1/4}$$ $$\frac{dr}{d\theta}=\theta^{-1/2}+\theta^{-1/3}+\theta^{-1/4}$$
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