Chapter 12 - Functions of Several Veriables - Review Exercises - Page 960: 45

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

$$\eqalign{ & f\left( {x,y,z} \right) = {e^{x + 2y + 3z}} \cr & {\text{Find the partial derivatives }}{f_x}\left( {x,y,z} \right){\text{,}}\,\,{f_y}\left( {x,y,z} \right){\text{ and }}{f_z}\left( {x,y} \right){\text{ }} \cr & {f_x}\left( {x,y,z} \right) = \frac{\partial }{{\partial x}}\left[ {{e^{x + 2y + 3z}}} \right] \cr & {\text{treat }}y{\text{ and }}z{\text{ as a constant}}{\text{, use the rule }}\left( {{e^u}} \right)' = {e^u}\left( {u'} \right) \cr & {f_x}\left( {x,y,z} \right) = {e^{x + 2y + 3z}}\frac{\partial }{{\partial x}}\left[ {x + 2y + 3z} \right] \cr & {\text{then}} \cr & {f_x}\left( {x,y,z} \right) = {e^{x + 2y + 3z}}\left( 1 \right) \cr & {f_x}\left( {x,y,z} \right) = {e^{x + 2y + 3z}} \cr & \cr & {f_y}\left( {x,y,z} \right) = \frac{\partial }{{\partial y}}\left[ {{e^{x + 2y + 3z}}} \right] \cr & {\text{treat }}y{\text{ and }}z{\text{ as a constant}}{\text{, use the rule }}\left( {{e^u}} \right)' = {e^u}\left( {u'} \right) \cr & {f_y}\left( {x,y,z} \right) = {e^{x + 2y + 3z}}\frac{\partial }{{\partial y}}\left[ {x + 2y + 3z} \right] \cr & {\text{then}} \cr & {f_y}\left( {x,y,z} \right) = {e^{x + 2y + 3z}}\left( 2 \right) \cr & {f_y}\left( {x,y,z} \right) = 2{e^{x + 2y + 3z}} \cr & \cr & and \cr & \cr & {f_z}\left( {x,y,z} \right) = \frac{\partial }{{\partial z}}\left[ {{e^{x + 2y + 3z}}} \right] \cr & {\text{treat }}y{\text{ and }}z{\text{ as a constant}}{\text{, use the rule }}\left( {{e^u}} \right)' = {e^u}\left( {u'} \right) \cr & {f_z}\left( {x,y,z} \right) = {e^{x + 2y + 3z}}\frac{\partial }{{\partial z}}\left[ {x + 2y + 3z} \right] \cr & {\text{then}} \cr & {f_z}\left( {x,y,z} \right) = {e^{x + 2y + 3z}}\left( 3 \right) \cr & {f_z}\left( {x,y,z} \right) = 3{e^{x + 2y + 3z}} \cr} $$