$${f}\left({x},{y}\right)=\frac{\left({x}+{y}\right)^{\mathrm{2}} }{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} } \\$$$$\begin{cases}{\frac{\partial{f}}{\partial{u}}=?}\\{\frac{\partial{f}}{\partial{v}}=?}\end{cases} \\$$$$\begin{cases}{{x}={uv}}\\{{y}={u}+{v}}\end{cases} \\$$
$${f}\left({u},{v}\right)=\mathrm{1}+\frac{\mathrm{2}{xy}}{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \:}\:=\:\mathrm{1}+\frac{\mathrm{2}{u}^{\mathrm{2}} {v}+\mathrm{2}{uv}^{\mathrm{2}} }{\left({uv}\right)^{\mathrm{2}} +\left({u}+{v}\right)^{\mathrm{2}} } \\$$$$\frac{\partial{f}}{\partial{u}}=\frac{\left[\left({uv}\right)^{\mathrm{2}} +\left({u}+{v}\right)^{\mathrm{2}} \right]\left[\mathrm{4}{uv}+\mathrm{2}{v}^{\mathrm{2}} \right]−\left(\mathrm{2}{u}^{\mathrm{2}} {v}+\mathrm{2}{uv}^{\mathrm{2}} \right)\left(\mathrm{2}{uv}+\mathrm{2}{u}+\mathrm{2}{v}\right)}{\left[\left({uv}\right)^{\mathrm{2}} +\left({u}+{v}\right)^{\mathrm{2}} \right]^{\mathrm{2}} } \\$$$${Simplification} \\$$$$\mathrm{4}{u}^{\mathrm{3}} {v}^{\mathrm{3}} +\mathrm{4}{u}^{\mathrm{3}} {v}+\mathrm{8}{u}^{\mathrm{2}} {v}^{\mathrm{2}} +\mathrm{4}{uv}^{\mathrm{3}} +\mathrm{2}{u}^{\mathrm{2}} {v}^{\mathrm{4}} +\mathrm{2}{u}^{\mathrm{2}} {v}^{\mathrm{2}} +\mathrm{4}{uv}^{\mathrm{3}} +\mathrm{2}{v}^{\mathrm{4}} \\$$$$−\mathrm{4}{u}^{\mathrm{3}} {v}^{\mathrm{2}} −\mathrm{4}{u}^{\mathrm{2}} {v}^{\mathrm{3}} −\mathrm{4}{u}^{\mathrm{3}} {v}−\mathrm{4}{u}^{\mathrm{2}} {v}^{\mathrm{2}} −\mathrm{4}{u}^{\mathrm{2}} {v}^{\mathrm{2}} −\mathrm{4}{uv}^{\mathrm{3}} \\$$$$=\mathrm{4}{u}^{\mathrm{3}} {v}^{\mathrm{3}} +\mathrm{2}{u}^{\mathrm{2}} {v}^{\mathrm{4}} +\mathrm{2}{u}^{\mathrm{2}} {v}^{\mathrm{2}} +\mathrm{4}{uv}^{\mathrm{3}} +\mathrm{2}{v}^{\mathrm{4}} −\mathrm{4}{u}^{\mathrm{3}} {v}^{\mathrm{2}} −\mathrm{4}{u}^{\mathrm{2}} {v}^{\mathrm{3}} \\$$$$\frac{\partial{f}}{\partial{u}}=\frac{\mathrm{4}{u}^{\mathrm{3}} {v}^{\mathrm{3}} +\mathrm{2}{u}^{\mathrm{2}} {v}^{\mathrm{4}} +\mathrm{2}{u}^{\mathrm{2}} {v}^{\mathrm{2}} +\mathrm{4}{uv}^{\mathrm{3}} +\mathrm{2}{v}^{\mathrm{4}} −\mathrm{4}{u}^{\mathrm{3}} {v}^{\mathrm{2}} −\mathrm{4}{u}^{\mathrm{2}} {v}^{\mathrm{3}} }{\left[\left({uv}\right)^{\mathrm{2}} +\left({u}+{v}\right)^{\mathrm{2}} \right]^{\mathrm{2}} } \\$$$$\frac{\partial{f}}{\partial{v}}=\frac{\mathrm{4}{u}^{\mathrm{3}} {v}^{\mathrm{3}} +\mathrm{2}{v}^{\mathrm{2}} {u}^{\mathrm{4}} +\mathrm{2}{u}^{\mathrm{2}} {v}^{\mathrm{2}} +\mathrm{4}{vu}^{\mathrm{3}} +\mathrm{2}{u}^{\mathrm{4}} −\mathrm{4}{v}^{\mathrm{3}} {u}^{\mathrm{2}} −\mathrm{4}{v}^{\mathrm{2}} {u}^{\mathrm{3}} }{\left[\left({uv}\right)^{\mathrm{2}} +\left({u}+{v}\right)^{\mathrm{2}} \right]^{\mathrm{2}} } \\$$