Chapter 12 - Functions of Several Veriables - Review Exercises - Page 960: 39

$${f_x}\left( {x,y} \right) = 6x{y^5}\,\,\,\,\,\,\,\,\,\,\,\,{f_y}\left( {x,y} \right) = 15{x^2}{y^4}$$

$$\eqalign{ & f\left( {x,y} \right) = 3{x^2}{y^5} \cr & {\text{Find the partial derivatives }}{f_x}\left( {x,y} \right){\text{ and }}{f_y}\left( {x,y} \right){\text{ then}} \cr & {f_x}\left( {x,y} \right) = \frac{\partial }{{\partial x}}\left[ {3{x^2}{y^5}} \right] \cr & {\text{treat }}y{\text{ as a constant}}{\text{,}} \cr & {f_x}\left( {x,y} \right) = {y^5}\frac{\partial }{{\partial x}}\left[ {3{x^2}} \right] \cr & {\text{solve the derivative}} \cr & {f_x}\left( {x,y} \right) = {y^5}\left( {6x} \right) \cr & {f_x}\left( {x,y} \right) = 6x{y^5} \cr & \cr & and \cr & \cr & {f_y}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left[ {3{x^2}{y^5}} \right] \cr & {\text{treat }}x{\text{ as a constant}}{\text{,}} \cr & {f_y}\left( {x,y} \right) = 3{x^2}\frac{\partial }{{\partial x}}\left[ {{y^5}} \right] \cr & {\text{solve the derivative}} \cr & {f_y}\left( {x,y} \right) = 3{x^2}\left( {5{y^4}} \right) \cr & {f_y}\left( {x,y} \right) = 15{x^2}{y^4} \cr} $$