Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 14: Partial Derivatives - Practice Exercises - Page 864: 31


$\frac{\partial w}{\partial r}=2$ $\frac{\partial w}{\partial s}=2-\pi$

Work Step by Step

$x=r+sin(s)=\pi$ when $r=\pi, s=0$ $\frac{dx}{dr}=1, \frac{dx}{ds}=cos(s)=1$ when $r=\pi, s=0$ $y=rs=0$ when $r=\pi, s=0$ $\frac{dy}{dr}=s=0, \frac{dy}{ds}=r=\pi$ when $r=\pi, s=0$ $w=sin(2x-y)=0$ when $r=\pi, s=0$ $\frac{dw}{dx}=2cos(2x-y)=2, \frac{dw}{dy}=-cos(2x-y)=-1$ when $r=\pi, s=0$ $\frac{\partial w}{\partial r}=\frac{dw}{dx}\frac{dx}{dr}+\frac{dw}{dy}\frac{dy}{dr}=2\times1-1\times0=2$ when $r=\pi, s=0$ $\frac{\partial w}{\partial s}=\frac{dw}{dx}\frac{dx}{ds}+\frac{dw}{dy}\frac{dy}{ds}=2\times1-1\times\pi=2-\pi$ when $r=\pi, s=0$
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