Chapter 12 - Functions of Several Veriables - 12.4 Partial Derivatives - 12.4 Exercises - Page 904: 38

$$\eqalign{ & {H_{xx}}\left( {x,y} \right) = {x^2}{\left( {4 + {x^2} + {y^2}} \right)^{ - 3/2}} + {\left( {4 + {x^2} + {y^2}} \right)^{ - 1/2}} \cr & {H_{yy}}\left( {x,y} \right) = {y^2}{\left( {4 + {x^2} + {y^2}} \right)^{ - 3/2}} + {\left( {4 + {x^2} + {y^2}} \right)^{ - 1/2}} \cr & {H_{xy}}\left( {x,y} \right) = {H_{yx}}\left( {x,y} \right) = xy{\left( {4 + {x^2} + {y^2}} \right)^{ - 3/2}} \cr} $$

$$\eqalign{ & H\left( {x,y} \right) = \sqrt {4 + {x^2} + {y^2}} \cr & H\left( {x,y} \right) = {\left( {4 + {x^2} + {y^2}} \right)^{1/2}} \cr & {\text{Find the first partial derivatives }}{H_x}\left( {x,y} \right){\text{ and }}{H_y}\left( {x,y} \right){\text{ then}} \cr & {H_x}\left( {x,y} \right) = \frac{\partial }{{\partial x}}\left[ {{{\left( {4 + {x^2} + {y^2}} \right)}^{1/2}}} \right] \cr & {\text{treat }}y{\text{ as a constant}}{\text{, then use chain rule}} \cr & {H_x}\left( {x,y} \right) = \frac{1}{2}{\left( {4 + {x^2} + {y^2}} \right)^{ - 1/2}}\frac{\partial }{{\partial x}}\left[ {4 + {x^2} + {y^2}} \right] \cr & {H_x}\left( {x,y} \right) = \frac{1}{2}{\left( {4 + {x^2} + {y^2}} \right)^{ - 1/2}}\left( {2x} \right) \cr & {H_x}\left( {x,y} \right) = x{\left( {4 + {x^2} + {y^2}} \right)^{ - 1/2}} \cr & and \cr & {H_y}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left[ {{{\left( {4 + {x^2} + {y^2}} \right)}^{1/2}}} \right] \cr & {\text{treat }}x{\text{ as a constant}}{\text{, then }} \cr & {H_y}\left( {x,y} \right) = \frac{1}{2}{\left( {4 + {x^2} + {y^2}} \right)^{ - 1/2}}\frac{\partial }{{\partial y}}\left[ {4 + {x^2} + {y^2}} \right] \cr & {H_y}\left( {x,y} \right) = \frac{1}{2}{\left( {4 + {x^2} + {y^2}} \right)^{ - 1/2}}\left( {2y} \right) \cr & {H_y}\left( {x,y} \right) = y{\left( {4 + {x^2} + {y^2}} \right)^{ - 1/2}} \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 & {H_{xy}}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left[ {x{{\left( {4 + {x^2} + {y^2}} \right)}^{ - 1/2}}} \right] \cr & {H_{xy}}\left( {x,y} \right) = x\frac{\partial }{{\partial y}}\left[ {{{\left( {4 + {x^2} + {y^2}} \right)}^{ - 1/2}}} \right] \cr & {H_{xy}}\left( {x,y} \right) = x\left[ {\frac{1}{2}{{\left( {4 + {x^2} + {y^2}} \right)}^{ - 3/2}}} \right]\frac{\partial }{{\partial y}}\left[ {4 + {x^2} + {y^2}} \right] \cr & {H_{xy}}\left( {x,y} \right) = x\left[ {\frac{1}{2}{{\left( {4 + {x^2} + {y^2}} \right)}^{ - 3/2}}} \right]\left( {2y} \right) \cr & {H_{xy}}\left( {x,y} \right) = xy{\left( {4 + {x^2} + {y^2}} \right)^{ - 3/2}} \cr & and \cr & {H_{yx}}\left( {x,y} \right) = \frac{\partial }{{\partial x}}\left[ {y{{\left( {4 + {x^2} + {y^2}} \right)}^{ - 1/2}}} \right] \cr & {H_{yx}}\left( {x,y} \right) = y\frac{\partial }{{\partial x}}\left[ {{{\left( {4 + {x^2} + {y^2}} \right)}^{ - 1/2}}} \right] \cr & {H_{yx}}\left( {x,y} \right) = y\left[ {\frac{1}{2}{{\left( {4 + {x^2} + {y^2}} \right)}^{ - 3/2}}} \right]\frac{\partial }{{\partial x}}\left[ {4 + {x^2} + {y^2}} \right] \cr & {H_{yx}}\left( {x,y} \right) = y\left[ {\frac{1}{2}{{\left( {4 + {x^2} + {y^2}} \right)}^{ - 3/2}}} \right]\left( {2x} \right) \cr & {H_{yx}}\left( {x,y} \right) = xy{\left( {4 + {x^2} + {y^2}} \right)^{ - 3/2}} \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 & {H_{xx}}\left( {x,y} \right) = \frac{\partial }{{\partial x}}\left[ {x{{\left( {4 + {x^2} + {y^2}} \right)}^{ - 1/2}}} \right] \cr & {\text{use product rule}} \cr & {H_{xx}}\left( {x,y} \right) = x\frac{\partial }{{\partial x}}\left[ {{{\left( {4 + {x^2} + {y^2}} \right)}^{ - 1/2}}} \right] + {\left( {4 + {x^2} + {y^2}} \right)^{ - 1/2}}\frac{\partial }{{\partial x}}\left[ x \right] \cr & {H_{xx}}\left( {x,y} \right) = x\left( { - \frac{1}{2}} \right){\left( {4 + {x^2} + {y^2}} \right)^{ - 3/2}}\left[ {2x} \right] + {\left( {4 + {x^2} + {y^2}} \right)^{ - 1/2}}\left( 1 \right) \cr & {H_{xx}}\left( {x,y} \right) = {x^2}{\left( {4 + {x^2} + {y^2}} \right)^{ - 3/2}} + {\left( {4 + {x^2} + {y^2}} \right)^{ - 1/2}} \cr & and \cr & {H_{yy}}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left[ {y{{\left( {4 + {x^2} + {y^2}} \right)}^{ - 1/2}}} \right] \cr & {\text{use product rule}} \cr & {H_{yy}}\left( {x,y} \right) = y\frac{\partial }{{\partial y}}\left[ {{{\left( {4 + {x^2} + {y^2}} \right)}^{ - 1/2}}} \right] + {\left( {4 + {x^2} + {y^2}} \right)^{ - 1/2}}\frac{\partial }{{\partial y}}\left[ y \right] \cr & {H_{yy}}\left( {x,y} \right) = y\left( { - \frac{1}{2}} \right){\left( {4 + {x^2} + {y^2}} \right)^{ - 3/2}}\left( {2y} \right) + {\left( {4 + {x^2} + {y^2}} \right)^{ - 1/2}}\left( 1 \right) \cr & {H_{yy}}\left( {x,y} \right) = {y^2}{\left( {4 + {x^2} + {y^2}} \right)^{ - 3/2}} + {\left( {4 + {x^2} + {y^2}} \right)^{ - 1/2}} \cr} $$