Chapter 12 - Functions of Several Veriables - 12.4 Partial Derivatives - 12.4 Exercises - Page 904: 33

$$\eqalign{ & {f_{xx}}\left( {x,y} \right) = - 16{y^3}\sin 4x,\,\,\,\,\,\,\,\,{\text{ }}{f_{yy}}\left( {x,y} \right) = 6y\sin 4x,\,\,\,\, \cr & {f_{xy}}\left( {x,y} \right) = 12{y^2}\cos 4x{\text{ and }}{f_{yx}}\left( {x,y} \right) = 12{y^2}\cos 4x \cr} $$

$$\eqalign{ & f\left( {x,y} \right) = {y^3}\sin 4x \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[ {{y^3}\sin 4x} \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[ {\sin 4x} \right] \cr & {f_x}\left( {x,y} \right) = {y^3}\left( {4\cos 4x} \right) \cr & {f_x}\left( {x,y} \right) = 4{y^3}\cos 4x \cr & and \cr & {f_y}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left[ {{y^3}\sin 4x} \right] \cr & {\text{treat }}x{\text{ as a constant}}{\text{, then }} \cr & {f_y}\left( {x,y} \right) = \sin 4x\frac{\partial }{{\partial y}}\left[ {{y^3}} \right] \cr & {f_y}\left( {x,y} \right) = \sin 4x\left( {3{y^2}} \right) \cr & {f_y}\left( {x,y} \right) = 3{y^2}\sin 4x \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[ {4{y^3}\cos 4x} \right] \cr & {f_{xy}}\left( {x,y} \right) = 4\cos 4x\frac{\partial }{{\partial y}}\left[ {{y^3}} \right] \cr & {f_{xy}}\left( {x,y} \right) = 12{y^2}\cos 4x \cr & and \cr & {\text{ }}{f_{yx}}\left( {x,y} \right) = \frac{\partial }{{\partial x}}\left[ {3{y^2}\sin 4x} \right] \cr & {\text{ }}{f_{yx}}\left( {x,y} \right) = 3{y^2}\frac{\partial }{{\partial x}}\left[ {\sin 4x} \right] \cr & {\text{ }}{f_{yx}}\left( {x,y} \right) = 3{y^2}\left( {4\cos 4x} \right) \cr & {\text{ }}{f_{yx}}\left( {x,y} \right) = 12{y^2}\cos 4x \cr & \cr & {\text{Find the second partial derivatives }}{f_{xx}}\left( {x,y} \right){\text{ and }}{f_{yy}}\left( {x,y} \right){\text{ then}} \cr & {f_{xx}}\left( {x,y} \right) = \frac{\partial }{{\partial x}}\left[ {4{y^3}\cos 4x} \right] \cr & {f_{xx}}\left( {x,y} \right) = 4{y^3}\frac{\partial }{{\partial x}}\left[ {\cos 4x} \right] \cr & {f_{xx}}\left( {x,y} \right) = 4{y^3}\left( { - 4\sin 4x} \right) \cr & {f_{xx}}\left( {x,y} \right) = - 16{y^3}\sin 4x \cr & and \cr & {\text{ }}{f_{yy}}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left[ {3{y^2}\sin 4x} \right] \cr & {\text{ }}{f_{yy}}\left( {x,y} \right) = 3\sin 4x\frac{\partial }{{\partial y}}\left[ {{y^2}} \right] \cr & {\text{ }}{f_{yy}}\left( {x,y} \right) = 6y\sin 4x \cr} $$