Chapter 12 - Functions of Several Veriables - Review Exercises - Page 960: 43

$${f_x}\left( {x,y} \right) = x{y^2}{e^{xy}} + y{e^{xy}}\,\,\,\,\,\,\,\,\,\,\,{f_y}\left( {x,y} \right) = {x^2}y{e^{xy}} + x{e^{xy}}$$

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