Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 15 - Differentiation in Several Variables - 15.6 The Chain Rule - Exercises - Page 809: 29


$$-\frac{x e^{x y}+1}{x \cos (x z)}$$

Work Step by Step

Given $$ \quad e^{x y}+\sin (x z)+y=0$$ Consider $$F(x,y,z )=e^{x y}+\sin (x z)+y=0 $$ Then \begin{align*} F_{y}&=x e^{x y}+1\\ F_{z}&=x \cos (x z) \end{align*} Then \begin{align*} \frac{\partial z}{\partial y}&=-\frac{F_{y}}{F_{z}}\\ &=-\frac{x e^{x y}+1}{x \cos (x z)} \end{align*}
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