Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 2 - Derivatives - Review - Exercises - Page 197: 43


$$y''=-\frac{5 x^{4}}{y^{11}}$$

Work Step by Step

Given $$x^6+y^6=1$$ Differentiate with respect to $x$ $$6x^5+6y^5y'=0\ \ \Rightarrow \ \ y'=\frac{-x^5}{y^5}$$ and \begin{align*} y^{\prime \prime}&=-\frac{y^{5}\left(5 x^{4}\right)-x^{5}\left(5 y^{4} y^{\prime}\right)}{\left(y^{5}\right)^{2}}\\ &=-\frac{5 x^{4} y^{4}\left[y-x\left(-x^{5} / y^{5}\right)\right]}{y^{10}}\\ &=-\frac{5 x^{4}\left[\left(y^{6}+x^{6}\right) / y^{5}\right]}{y^{6}}\\ &=-\frac{5 x^{4}}{y^{11}} \end{align*}
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