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: 33

Answer

$$\frac{\partial}{\partial x}w=\frac{1}{x+2y+3z};$$ $$\frac{\partial}{\partial y}w=\frac{2}{x+2y+3z};$$ $$\frac{\partial}{\partial z}w=\frac{3}{x+2y+3z}.$$

Work Step by Step

Regard $y$ and $z$ as constant and differentiate with respect to $x$: $$\frac{\partial}{\partial x}w=\frac{\partial}{\partial x}\ln(x+2y+3z)=\frac{1}{x+2y+3z}\frac{\partial}{\partial x}(x+2y+3z)=\frac{1}{x+2y+3z}\cdot(1+0+0)=\frac{1}{x+2y+3z}.$$ Regard $x$ and $z$ as constant and differentiate with respect to $y$: $$\frac{\partial}{\partial y}w=\frac{\partial}{\partial y}\ln(x+2y+3z)=\frac{1}{x+2y+3z}\frac{\partial}{\partial y}(x+2y+3z)=\frac{1}{x+2y+3z}\cdot(0+2+0)=\frac{2}{x+2y+3z}.$$ Regard $x$ and $y$ as constant and differentiate with respect to $z$: $$\frac{\partial}{\partial z}w=\frac{\partial}{\partial z}\ln(x+2y+3z)=\frac{1}{x+2y+3z}\frac{\partial}{\partial z}(x+2y+3z)=\frac{1}{x+2y+3z}\cdot(0+0+3)=\frac{3}{x+2y+3z}.$$
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