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: 31

Answer

$$\frac{\partial}{\partial x}f(x,y)=3x^2yz^2;$$ $$\frac{\partial}{\partial y}f(x,y)=x^3z^2+2z;$$ $$\frac{\partial}{\partial z}f(x,y)=2x^3yz+2y.$$

Work Step by Step

Regard $y$ and $z$ as constant and differentiate with respect to $x$: $$\frac{\partial}{\partial x}f(x,y)=\frac{\partial}{\partial x}(x^3yz^2+2yz)=yz^2\cdot3x^2+0=3x^2yz^2.$$ Regard $x$ and $z$ as constant and differentiate with respect to $y$: $$\frac{\partial}{\partial y}f(x,y)=\frac{\partial}{\partial y}(x^3yz^2+2yz)=x^3z^2\cdot 1+2z\cdot 1=x^3z^2+2z.$$ Regard $x$ and $y$ as constant and differentiate with respect to $z$: $$\frac{\partial}{\partial z}f(x,y)=\frac{\partial}{\partial z}(x^3yz^2+2yz)=x^3y\cdot 2z+2y\cdot1=2x^3yz+2y.$$
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