Calculus with Applications (10th Edition)

Published by Pearson
ISBN 10: 0321749006
ISBN 13: 978-0-32174-900-0

Chapter 9 - Multiveriable Calculus - Chapter Review - Review Exercises - Page 518: 33


$${f_x}\left( {x,y} \right) = 3{x^2}{e^{3y}}{\text{ and }}{f_y}\left( {x,y} \right) = 3{x^3}{e^{3y}}$$

Work Step by Step

$$\eqalign{ & f\left( {x,y} \right) = {x^3}{e^{3y}} \cr & {\text{treat y as a constant and }}x{\text{ as a variable}}{\text{. then}} \cr & {f_x}\left( {x,y} \right) = \frac{\partial }{{\partial x}}\left[ {{x^3}{e^{3y}}} \right] \cr & {f_x}\left( {x,y} \right) = {e^{3y}}\frac{\partial }{{\partial x}}\left[ {{x^3}} \right] \cr & {f_x}\left( {x,y} \right) = {e^{3y}}\left( {3{x^2}} \right) \cr & {f_x}\left( {x,y} \right) = 3{x^2}{e^{3y}} \cr & \cr & {f_y}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left[ {{x^3}{e^{3y}}} \right] \cr & {f_y}\left( {x,y} \right) = {x^3}\frac{\partial }{{\partial y}}\left[ {{e^{3y}}} \right] \cr & {\text{solving derivatives}} \cr & {f_y}\left( {x,y} \right) = {x^3}\left( {3{e^{3y}}} \right) \cr & {f_y}\left( {x,y} \right) = 3{x^3}{e^{3y}} \cr} $$
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