$$ f(x,y)=\frac{x \cdot y^2}{x^2+y^2} $$
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$$ \frac {\partial f(x,y)}{\partial x}=\frac{y^2 \cdot (x^2+y^2)-x \cdot y^2 \cdot 2x}{(x^2+y^2)^2} $$
$$ \frac {\partial f(x,y)}{\partial x}=\frac{y^2 \cdot (x^2+y^2-2x^2 )}{(x^2+y^2)^2} $$
$$ \frac {\partial f(x,y)}{\partial x}=\frac{y^2 \cdot (y^2-x^2 )}{(x^2+y^2)^2} $$
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$$ \frac {\partial f(x,y)}{\partial y}=\frac{2xy( x^2+y^2) - x \cdot y^2 \cdot 2y }{( x^2+y^2) ^2} $$
$$ \frac {\partial f(x,y)}{\partial y}=\frac{2xy( x^2+y^2 - y^2 ) }{( x^2+y^2) ^2} $$
$$ \frac {\partial f(x,y)}{\partial y}=\frac{2x^3y }{( x^2+y^2) ^2} $$