Chapter 12 - Functions of Several Veriables - 12.4 Partial Derivatives - 12.4 Exercises - Page 904: 31

$${f_{xx}}\left( {x,y} \right) = 2{y^3},\,\,\,\,\,\,\,\,{\text{ }}{f_{yy}}\left( {x,y} \right) = 6{x^2}y,\,\,\,\,{f_{xy}}\left( {x,y} \right) = 6x{y^2}{\text{ and }}{f_{yx}}\left( {x,y} \right) = 6x{y^2}$$

$$\eqalign{ & f\left( {x,y} \right) = {x^2}{y^3} \cr & {\text{Find the first partial derivatives }}{f_x}\left( {x,y} \right){\text{ and }}{f_y}\left( {x,y} \right){\text{ then}} \cr & {f_x}\left( {x,y} \right) = \frac{\partial }{{\partial x}}\left[ {{x^2}{y^3}} \right] \cr & {\text{treat }}y{\text{ as a constant}}{\text{, then}} \cr & {f_x}\left( {x,y} \right) = {y^3}\frac{\partial }{{\partial x}}\left[ {{x^2}} \right] \cr & {f_x}\left( {x,y} \right) = 2x{y^3} \cr & and \cr & {f_y}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left[ {{x^2}{y^3}} \right] \cr & {\text{treat }}x{\text{ as a constant}}{\text{, then }} \cr & {f_y}\left( {x,y} \right) = {x^2}\frac{\partial }{{\partial y}}\left[ {{y^3}} \right] \cr & {f_y}\left( {x,y} \right) = 3{x^2}{y^2} \cr & \cr & {\text{Find the second partial derivatives }}{f_{xy}}\left( {x,y} \right){\text{ and }}{f_{yx}}\left( {x,y} \right){\text{ then}} \cr & {f_{xy}}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left[ {2x{y^3}} \right] \cr & {f_{xy}}\left( {x,y} \right) = 6x{y^2} \cr & and \cr & {\text{ }}{f_{yx}}\left( {x,y} \right) = \frac{\partial }{{\partial x}}\left[ {3{x^2}{y^2}} \right] \cr & {\text{ }}{f_{yx}}\left( {x,y} \right) = 3{y^2}\frac{\partial }{{\partial x}}\left[ {{x^2}} \right] \cr & {\text{ }}{f_{yx}}\left( {x,y} \right) = 6x{y^2} \cr & \cr & {\text{Find the second partial derivatives }}{h_{xx}}\left( {x,y} \right){\text{ and }}{h_{yy}}\left( {x,y} \right){\text{ then}} \cr & {f_{xx}}\left( {x,y} \right) = \frac{\partial }{{\partial x}}\left[ {2x{y^3}} \right] \cr & {f_{xx}}\left( {x,y} \right) = 2{y^3}\frac{\partial }{{\partial x}}\left[ x \right] \cr & {f_{xx}}\left( {x,y} \right) = 2{y^3} \cr & and \cr & {\text{ }}{f_{yy}}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left[ {3{x^2}{y^2}} \right] \cr & {\text{ }}{f_{yy}}\left( {x,y} \right) = 6{x^2}y \cr} $$