## Calculus with Applications (10th Edition)

$${f_x}\left( {x,y} \right) = - 12{x^3}{y^3},\,\,\,\,{f_y}\left( {x,y} \right) = - 9{x^4}{y^2},\,\,\,\,\,{f_x}\left( {2, - 1} \right) = 96{\text{ and }}{f_y}\left( { - 4,3} \right) = - 20,736$$
\eqalign{ & f\left( {x,y} \right) = - 3{x^4}{y^3} + 10 \cr & {\text{Find }}{f_x}\left( {x,y} \right){\text{ and }}{f_y}\left( {x,y} \right) \cr & \cr & {f_x}\left( {x,y} \right) = \frac{\partial }{{\partial x}}\left[ { - 3{x^4}{y^3} + 10} \right] \cr & {\text{treat y as a constant and }}x{\text{ as a variable}}{\text{. then}} \cr & {f_x}\left( {x,y} \right) = \frac{\partial }{{\partial x}}\left[ { - 3{x^4}{y^3}} \right] + \frac{\partial }{{\partial x}}\left[ {10} \right] \cr & {f_x}\left( {x,y} \right) = - 3{y^3}\frac{\partial }{{\partial x}}\left[ {{x^4}} \right] + \frac{\partial }{{\partial x}}\left[ {10} \right] \cr & {f_x}\left( {x,y} \right) = - 3{y^3}\left( {4{x^3}} \right) \cr & {f_x}\left( {x,y} \right) = - 12{x^3}{y^3} \cr & {\text{evaluate }}{f_x}\left( {2, - 1} \right) \cr & {f_x}\left( {2, - 1} \right) = - 12{\left( 2 \right)^3}{\left( { - 1} \right)^3} \cr & {f_x}\left( {2, - 1} \right) = - 12\left( 8 \right)\left( { - 1} \right) \cr & {f_x}\left( {2, - 1} \right) = 96 \cr & \cr & {f_y}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left[ { - 3{x^4}{y^3} + 10} \right] \cr & {\text{treat x as a constant and y as a variable}}{\text{. then}} \cr & {f_y}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left[ { - 3{x^4}{y^3}} \right] + \frac{\partial }{{\partial y}}\left[ {10} \right] \cr & {f_y}\left( {x,y} \right) = - 3{x^4}\frac{\partial }{{\partial y}}\left[ {{y^3}} \right] + \frac{\partial }{{\partial y}}\left[ {10} \right] \cr & {f_y}\left( {x,y} \right) = - 3{x^4}\left( {3{y^2}} \right) \cr & {f_y}\left( {x,y} \right) = - 9{x^4}{y^2} \cr & {\text{evaluate }}{f_y}\left( { - 4,3} \right) \cr & {f_y}\left( { - 4,3} \right) = - 9{\left( { - 4} \right)^4}{\left( 3 \right)^2} \cr & {f_y}\left( { - 4,3} \right) = - 9\left( {256} \right)\left( 9 \right) \cr & {f_y}\left( { - 4,3} \right) = - 20,736 \cr}