Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 13 - Functions of Several Variables - 13.3 Exercises - Page 896: 17

Answer

$$\frac{\partial z}{\partial x}=2xe^{2y}$$ $$\frac{\partial z}{\partial y}=2x^2e^{2y}$$

Work Step by Step

The partial derivative with respect to $x$ is: $$\frac{\partial z}{\partial x}=\frac{\partial }{\partial x}(x^2e^{2y})=e^{2y}\frac{\partial }{\partial x}(x^2)=2xe^{2y}$$ Here $y$ is treated as a constant, so $e^{2y}$ is treated as a constant as well. The partial derivative with respect to $y$ is: $$\frac{\partial z}{\partial y}=\frac{\partial }{\partial y}(x^2e^{2y})=x^2\frac{\partial }{\partial y}(e^{2y})=x^2e^{2y}\frac{\partial }{\partial y}(2y)=2x^2e^{2y}$$ Here $x$ is treated as a constant, so $x^2$ is treated as a constant as well. Also, we used the chain rule to find $\frac{\partial }{\partial y}(e^{2y}).$
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