## Thomas' Calculus 13th Edition

Published by Pearson

# Chapter 14: Partial Derivatives - Practice Exercises - Page 864: 30

5

#### Work Step by Step

$x=2\sqrt t=2$ (t=1) $\frac{dx}{dt}=\frac{1}{\sqrt t}=1$ (t=1) $y=t-1+\ln t=0$ (t=1) $\frac{dy}{dt}=1+\frac{1}{t}=2$ (t=1) $z=\pi t=\pi$ (t=1) $\frac{dz}{dt}=\pi$ $w=xe^{y}+ySin(z)-Cos(z)$ $w_{x}=e^{y}=1$ (t=1) $w_{y}=xe^{y}+Sin(z)=2$ (t=1) $w_{z}=yCos(z)+Sin(z)=0$ (t=1) $\frac{dw}{dt}=w_{x}\frac{dx}{dt}+w_{y}\frac{dy}{dt}+w_{z}\frac{dz}{dt}=1+4+0=5$ (t=1)

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