Chapter 12 - Functions of Several Veriables - 12.4 Partial Derivatives - 12.4 Exercises - Page 904: 30

$$\eqalign{ & {f_{xx}}\left( {x,y} \right) = 40{x^3}{y^2} + 2y,\,\,\,\,\,\,\,\,{\text{ }}{f_{yy}}\left( {x,y} \right) = 4{x^5},\,\,\,\,\,\,\, \cr & {f_{xy}}\left( {x,y} \right) = 20{x^4}y + 2x{\text{ and }}{f_{yx}}\left( {x,y} \right) = 20{x^4}y + 2x \cr} $$

$$\eqalign{ & f\left( {x,y} \right) = 2{x^5}{y^2} + {x^2}y \cr & {\text{Find the first partial derivatives }}{f_x}\left( {x,y} \right){\text{ and }}{f_y}\left( {x,y} \right){\text{ then}} \cr & {f_x}\left( {x,y} \right) = \frac{\partial }{{\partial x}}\left[ {2{x^5}{y^2} + {x^2}y} \right] \cr & {\text{treat }}y{\text{ as a constant}}{\text{, then}} \cr & {f_x}\left( {x,y} \right) = 10{x^4}{y^2} + 2xy \cr & and \cr & {f_y}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left[ {2{x^5}{y^2} + {x^2}y} \right] \cr & {\text{treat }}x{\text{ as a constant}}{\text{, then }} \cr & {f_y}\left( {x,y} \right) = 4{x^5}y + {x^2} \cr & \cr & {\text{Find the second partial derivatives }}{f_{xy}}\left( {x,y} \right){\text{ and }}{f_{yx}}\left( {x,y} \right){\text{ then}} \cr & {f_{xy}}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left[ {10{x^4}{y^2} + 2xy} \right] \cr & {f_{xy}}\left( {x,y} \right) = 20{x^4}y + 2x \cr & and \cr & {\text{ }}{f_{yx}}\left( {x,y} \right) = \frac{\partial }{{\partial x}}\left[ {4{x^5}y + {x^2}} \right] \cr & {\text{ }}{f_{yx}}\left( {x,y} \right) = 20{x^4}y + 2x \cr & \cr & {\text{Find the second partial derivatives }}{h_{xx}}\left( {x,y} \right){\text{ and }}{h_{yy}}\left( {x,y} \right){\text{ then}} \cr & {f_{xx}}\left( {x,y} \right) = \frac{\partial }{{\partial x}}\left[ {10{x^4}{y^2} + 2xy} \right] \cr & {f_{xx}}\left( {x,y} \right) = 40{x^3}{y^2} + 2y \cr & and \cr & {\text{ }}{f_{yy}}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left[ {4{x^5}y + {x^2}} \right] \cr & {\text{ }}{f_{yy}}\left( {x,y} \right) = 4{x^5} \cr} $$