Calculus 8th Edition

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

Chapter 14 - Partial Derivatives - 14.3 Partial Derivatives - 14.3 Exercises - Page 964: 25


The solution is $$\frac{\partial}{\partial u}g(u,v)=10uv(u^2v-v^3)^4;$$ $$\frac{\partial}{\partial v}g(u,v)=5(u^2v-v^3)^4(u^2-3v^2)$$

Work Step by Step

Regard $v$ and as a constant and differentiate with respect to $u$: $$\frac{\partial}{\partial u}g(u,v)=\frac{\partial}{\partial u}\bigg((u^2v-v^3)^5\bigg)=5(u^2v-v^3)^4(u^2v-v^3)^{'}_u=5(u^2v-v^3)^4\cdot2uv=10uv(u^2v-v^3)^4.$$ Regard $u$ as constand and differentiate with respect to $v$: $$\frac{\partial}{\partial v}g(u,v)=\frac{\partial}{\partial v}\bigg((u^2v-v^3)^5\bigg)=5(u^2v-v^3)^4(u^2v-v^3)^{'}_u=5(u^2v-v^3)^4(u^2-3v^2).$$
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