∂f/∂x = (-x^2+y^2)/(x^2+y^2)^2
∂^2f/∂x^2 = [(-x^2+y^2)'(x^2+y^2)^2-(-x^2+y^2){(x^2+y^2)^2}']/(x^2+y^2)^4
={(-2x)(x^2+y^2)^2-(-x^2+y^2)(4x)(x^2+y^2)}/(x^2+y^2)^4
={(-2x)(x^2+y^2)-4x(-x^2+y^2)}/(x^2+y^2)
= (-2x^3-2xy^2+4x^3-4xy^2)/(x^2+y^2)^4
=(2x^3-6xy^2)/(x^2+y^2)

∂f/∂y = -2xy/(x^2+y^2)^2
∂^2f/∂y^2 = {(-2x)(x^2+y^2)^2-(-2xy)(4y)(x^2+y^2)}/(x^2+y^2)^4
= {(-2x)(x^2+y^2)+8xy^2}/(x^2+y^2)^3
= (-2x^3+6xy^2)/(x^2+y^2)^3