Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 15 - Differentiation in Several Variables - 15.6 The Chain Rule - Exercises - Page 808: 14

Answer

$$244$$

Work Step by Step

Given $$g(x, y)=x^{2}-y^{2},\ \ \ x=s^{2}+1,\ \ y=1-2s$$ Since at $s=4$, $(x,y)= (17,-7)$ $$ \frac{\partial g}{\partial \:x}=2x,\ \ \ \ \ \frac{\partial g}{\partial \:y}=-2y \\ \frac{\partial x}{\partial s}=2s,\ \ \ \ \ \ \ \ \ \ \ \frac{\partial y}{\partial s}=-2 $$ Then \begin{align*} \frac{\partial g }{ \partial s}&=\frac{\partial g }{ \partial x}\frac{\partial x }{ \partial s}+\frac{\partial g }{ \partial y}\frac{\partial y }{ \partial s}\\ &= 4sx+4y \end{align*} Hence \begin{align*} \frac{\partial g }{ \partial s}\bigg|_{s=4}&= 4(4)(17)+4(-7)\\ &=244 \end{align*}
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