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 - Page 167: 30



Work Step by Step

$x^{\frac{2}{3}}+y^{\frac{2}{3}}=4\\ \frac{2}{3}x^{-1/3}+\frac{2}{3}y^{-1/3}y'=0\\ \frac{2}{3\sqrt[3] x}+\frac{2y'}{3\sqrt[3] y}=0\\ -\frac{2}{3\sqrt[3] x}=\frac{2y'}{3\sqrt[3] y}\\ y'=-\sqrt[3]{\frac{y}{x}}$ Plug in $(-3\sqrt 3,1)$ in $y'$ to find the gradient $y'=-\sqrt[3]{\frac{1}{-3\sqrt3}}\\ y'={\frac{1}{\sqrt[3]{3\sqrt3}}}$ $y'=\frac{1}{\sqrt[2]{3}}$ Equation of the tangent line at $(-3\sqrt 3,1)$ $y-1={\frac{1}{\sqrt[2]{3}}}(x-(-3\sqrt3))\\ y={\frac{1}{\sqrt[2]{3}}}x+4$
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