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: 27



Work Step by Step

Given $$x^{2} y+y^{2} z+x z^{2}=10$$ Consider $$F(x,y,z)=x^{2} y+y^{2} z+x z^{2}=10 $$ Then \begin{align*} F_x&= 2xy+z^2\\ F_z&= y^2+2xz \end{align*} Then \begin{align*} \frac{\partial z}{\partial x}&=-\frac{F_{x}}{F_{z}}\\ &=-\frac{2xy+z^2}{y^2+2xz} \end{align*}
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