\partial_{x}v=2x+\frac{4xy}{(2x^2+2y^2)^2}

\partial_{y}v=-2y-\frac{x^2-y^2}{2(x^2+y^2)^2}

\partial^2_{x}v=2-\frac{3x^2y+y^3}{(x^2+y^2)^3}

\partial^2_{y}v=-2+\frac{3x^2y+y^3}{(x^2+y^2)^3}