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: 37

Answer

$$\frac{\partial}{\partial x}h(x,y,z,t)=2xy\cos\frac{z}{t}.$$ $$\frac{\partial}{\partial y}h(x,y,z,t)=x^2\cos(z/t).$$ $$\frac{\partial}{\partial z}h(x,y,z,t)=-\frac{x^2y}{t}\sin\frac{z}{t}.$$ $$\frac{\partial}{\partial t}h(x,y,z,t)=\frac{x^2yz}{t^2}\sin\frac{z}{t}.$$

Work Step by Step

Regard $y,z$ and $t$ as constant and differentiate with respect to $x$: $$\frac{\partial}{\partial x}h(x,y,z,t)=\frac{\partial}{\partial x}(x^2y\cos(z/t))=y\cos(z/t)\frac{\partial}{\partial x}x^2=y\cos(z/t)\cdot 2x=2xy\cos\frac{z}{t}.$$ Regard $x,z$ and $t$ as constant and differentiate with respect to $y$: $$\frac{\partial}{\partial y}h(x,y,z,t)=\frac{\partial}{\partial y}(x^2y\cos(z/t))=x^2\cos(z/t)\frac{\partial}{\partial y}y=x^2\cos(z/t).$$ Regard $x,y$ and $t$ as constant and differentiate with respect to $z$: $$\frac{\partial}{\partial z}h(x,y,z,t)=\frac{\partial}{\partial z}(x^2y\cos(z/t))=x^2y\frac{\partial}{\partial z}\cos(z/t)=x^2y(-\sin(z/t))\frac{\partial}{\partial z}\frac{z}{t}=-x^2y\sin(z/t)\frac{1}{t}\frac{\partial}{\partial z}z=-\frac{x^2y}{t}\sin\frac{z}{t}.$$ Regard $x,y$ and $z$ as constant and differentiate with respect to $t$: $$\frac{\partial}{\partial t}h(x,y,z,t)=\frac{\partial}{\partial t}(x^2y\cos(z/t))=x^2y\frac{\partial}{\partial t}\cos(z/t)=x^2y(-\sin(z/t))\frac{\partial}{\partial t}\frac{z}{t}=-x^2y\sin(z/t)z\frac{\partial}{\partial t}\frac{1}{t}=-x^2yz\sin(z/t)\left(-\frac{1}{t^2}\right)=\frac{x^2yz}{t^2}\sin\frac{z}{t}.$$
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