staying with parametric functions makes it hard.
So, let
r^2 = x^2+y^2
tanθ = y/x
z = r^2 tan^2θ = (x^2+y^2)(y^2/x^2) = y^2 + y^4/x^2
∂z/∂x = -2y^4/x^3
∂z/∂y = 2y + 4y^3/x^2
∂^2z/∂x∂y = ∂^2z/∂y∂x = -8y^3/x^3 = -8tan^3θ
If z=r^2 tan^2(theta),x=r cos(theta),y=r sin(theta),
find the {partial^2 z}/{partial(x)partial(y)}
1 answer