Chapter 12 - Functions of Several Veriables - Review Exercises - Page 960: 41

$${f_x}\left( {x,y} \right) = \frac{{2x{y^2}}}{{{{\left( {{x^2} + {y^2}} \right)}^2}}}\,\,\,\,\,\,\,\,\,\,\,{f_y}\left( {x,y} \right) = - \frac{{2{x^2}y}}{{{{\left( {{x^2} + {y^2}} \right)}^2}}}$$

$$\eqalign{ & f\left( {x,y} \right) = \frac{{{x^2}}}{{{x^2} + {y^2}}} \cr & {\text{Find the partial derivatives }}{f_x}\left( {x,y} \right){\text{ and }}{f_y}\left( {x,y} \right){\text{ then}} \cr & {f_x}\left( {x,y} \right) = \frac{\partial }{{\partial x}}\left[ {\frac{{{x^2}}}{{{x^2} + {y^2}}}} \right] \cr & {\text{treat }}y{\text{ as a constant}}{\text{, use quotient rule}} \cr & {f_x}\left( {x,y} \right) = \frac{{\left( {{x^2} + {y^2}} \right)\left( {2x} \right) - {x^2}\left( {2x} \right)}}{{{{\left( {{x^2} + {y^2}} \right)}^2}}} \cr & {\text{simplifying}} \cr & {f_x}\left( {x,y} \right) = \frac{{2{x^3} + 2x{y^2} - 2{x^3}}}{{{{\left( {{x^2} + {y^2}} \right)}^2}}} \cr & {f_x}\left( {x,y} \right) = \frac{{2x{y^2}}}{{{{\left( {{x^2} + {y^2}} \right)}^2}}} \cr & \cr & and \cr & \cr & {f_y}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left[ {\frac{{{x^2}}}{{{x^2} + {y^2}}}} \right] \cr & {\text{treat }}x{\text{ as a constant}}{\text{,}} \cr & {f_y}\left( {x,y} \right) = {x^2}\frac{\partial }{{\partial y}}\left[ {\frac{1}{{{x^2} + {y^2}}}} \right] \cr & {f_y}\left( {x,y} \right) = {x^2}\frac{\partial }{{\partial y}}\left[ {{{\left( {{x^2} + {y^2}} \right)}^{ - 1}}} \right] \cr & {\text{by the quotient rule}} \cr & {f_y}\left( {x,y} \right) = - {x^2}{\left( {{x^2} + {y^2}} \right)^{ - 2}}\frac{\partial }{{\partial y}}\left[ {{x^2} + {y^2}} \right] \cr & {\text{solve the derivative}} \cr & {f_y}\left( {x,y} \right) = - {x^2}{\left( {{x^2} + {y^2}} \right)^{ - 2}}\left( {2y} \right) \cr & {f_y}\left( {x,y} \right) = - \frac{{2{x^2}y}}{{{{\left( {{x^2} + {y^2}} \right)}^2}}} \cr} $$