Chapter 9 - Multiveriable Calculus - 9.2 Partial Derivatives - 9.2 Exercises - Page 478: 26

$$\eqalign{ & {k_{xx}}\left( {x,y} \right) = \frac{{84y}}{{{{\left( {2x + 3y} \right)}^3}}},\,\,\,{k_{yy}}\left( {x,y} \right) = - \frac{{126x}}{{{{\left( {2x + 3y} \right)}^3}}} \cr & {k_{xy}}\left( {x,y} \right) = {k_{yx}}\left( {x,y} \right) = \frac{{63y - 42x}}{{{{\left( {2x + 3y} \right)}^3}}} \cr} $$

$$\eqalign{ & k\left( {x,y} \right) = \frac{{ - 7x}}{{2x + 3y}} \cr & {\text{Find }}{k_x}\left( {x,y} \right){\text{ and }}{k_y}\left( {x,y} \right) \cr & {k_x}\left( {x,y} \right) = \frac{\partial }{{\partial x}}\left[ {\frac{{ - 7x}}{{2x + 3y}}} \right] \cr & {\text{treat }}y{\text{ as a constant and }}x{\text{ as a variable}}{\text{. then by using quotient rule}} \cr & {k_x}\left( {x,y} \right) = \frac{{\left( {2x + 3y} \right)\left( { - 7} \right) + 7x\left( 2 \right)}}{{{{\left( {2x + 3y} \right)}^2}}} \cr & {k_x}\left( {x,y} \right) = \frac{{ - 14x - 21y + 14x}}{{{{\left( {2x + 3y} \right)}^2}}} \cr & {k_x}\left( {x,y} \right) = \frac{{ - 21y}}{{{{\left( {2x + 3y} \right)}^2}}} \cr & \cr & {k_y}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left[ {\frac{{ - 7x}}{{2x + 3y}}} \right] \cr & {\text{treat }}x{\text{ as a constant and }}y{\text{ as a variable}}{\text{. then }} \cr & {k_y}\left( {x,y} \right) = - 7x\frac{\partial }{{\partial y}}\left[ {\frac{1}{{2x + 3y}}} \right] \cr & {k_y}\left( {x,y} \right) = - 7x\left( { - \frac{1}{{{{\left( {2x + 3y} \right)}^2}}}} \right)\frac{\partial }{{\partial y}}\left[ {2x + 3y} \right] \cr & {k_y}\left( {x,y} \right) = - 7x\left( { - \frac{1}{{{{\left( {2x + 3y} \right)}^2}}}} \right)\left( 3 \right) \cr & {k_y}\left( {x,y} \right) = \frac{{21x}}{{{{\left( {2x + 3y} \right)}^2}}} \cr & \cr & {\text{find the second - order partial derivatives}} \cr & {k_{xy}}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left[ {\frac{{ - 21y}}{{{{\left( {2x + 3y} \right)}^2}}}} \right] \cr & {\text{by using the quotient rule}} \cr & {k_{xy}}\left( {x,y} \right) = \frac{{{{\left( {2x + 3y} \right)}^2}\left( { - 21} \right) + 21y\left( 2 \right)\left( {2x + 3y} \right)\left( 3 \right)}}{{{{\left( {2x + 3y} \right)}^4}}} \cr & {\text{simplifying}} \cr & {k_{xy}}\left( {x,y} \right) = \frac{{\left( {2x + 3y} \right)\left( { - 21} \right) + 21y\left( 2 \right)\left( 3 \right)}}{{{{\left( {2x + 3y} \right)}^3}}} \cr & {k_{xy}}\left( {x,y} \right) = \frac{{ - 42x - 63y + 126y}}{{{{\left( {2x + 3y} \right)}^3}}} \cr & {k_{xy}}\left( {x,y} \right) = \frac{{63y - 42x}}{{{{\left( {2x + 3y} \right)}^3}}} \cr & \cr & {k_{yx}}\left( {x,y} \right) = \frac{\partial }{{\partial x}}\left[ {\frac{{21x}}{{{{\left( {2x + 3y} \right)}^2}}}} \right] \cr & {\text{by using quotient rule}} \cr & {k_{yx}}\left( {x,y} \right) = \frac{{21{{\left( {2x + 3y} \right)}^2} - 21x\left( 2 \right)\left( {2x + 3y} \right)\left( 2 \right)}}{{{{\left( {2x + 3y} \right)}^4}}} \cr & {k_{yx}}\left( {x,y} \right) = \frac{{21\left( {2x + 3y} \right) - 21x\left( 2 \right)\left( 2 \right)}}{{{{\left( {2x + 3y} \right)}^3}}} \cr & {k_{yx}}\left( {x,y} \right) = \frac{{42x + 63y - 84x}}{{{{\left( {2x + 3y} \right)}^3}}} \cr & {k_{yx}}\left( {x,y} \right) = \frac{{63y - 42x}}{{{{\left( {2x + 3y} \right)}^3}}} \cr & \cr & {k_{xx}}\left( {x,y} \right) = \frac{\partial }{{\partial x}}\left[ {\frac{{ - 21y}}{{{{\left( {2x + 3y} \right)}^2}}}} \right] \cr & {k_{xx}}\left( {x,y} \right) = - 21y\frac{\partial }{{\partial x}}\left[ {{{\left( {2x + 3y} \right)}^{ - 2}}} \right] \cr & {k_{xx}}\left( {x,y} \right) = - 21y\left( { - 2} \right){\left( {2x + 3y} \right)^{ - 3}}\left( 2 \right) \cr & {k_{xx}}\left( {x,y} \right) = 84y{\left( {2x + 3y} \right)^{ - 3}} \cr & {k_{xx}}\left( {x,y} \right) = \frac{{84y}}{{{{\left( {2x + 3y} \right)}^3}}} \cr & \cr & {\text{and}} \cr & {k_{yy}}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left[ {\frac{{21x}}{{{{\left( {2x + 3y} \right)}^2}}}} \right] \cr & {k_{yy}}\left( {x,y} \right) = 21x\frac{\partial }{{\partial y}}\left[ {{{\left( {2x + 3y} \right)}^{ - 2}}} \right] \cr & {k_{yy}}\left( {x,y} \right) = 21x\left( { - 2} \right){\left( {2x + 3y} \right)^{ - 3}}\left( 3 \right) \cr & {k_{yy}}\left( {x,y} \right) = - 126x{\left( {2x + 3y} \right)^{ - 3}} \cr & {k_{yy}}\left( {x,y} \right) = - \frac{{126x}}{{{{\left( {2x + 3y} \right)}^3}}} \cr} $$