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: 28


$$-\frac{ 2wz+3y}{x^2+3w^2+z^2}$$

Work Step by Step

Given $$ \quad x^{2} w+w^{3}+w z^{2}+3 y z=0$$ Consider $$F(x,y,z,w)=x^{2} w+w^{3}+w z^{2}+3 y z=0 $$ Then \begin{align*} F_w&= x^2+3w^2+z^2\\ F_z&= 2wz+3y \end{align*} Then \begin{align*} \frac{\partial w}{\partial z}&=-\frac{F_{z}}{F_{w}}\\ &=-\frac{ 2wz+3y}{x^2+3w^2+z^2} \end{align*}
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