I note that there are three variables but f has only two arguments, so f(u,v)=f(z-xy,x²+y²) where u(x,y,z)=z-xy and v(x,y)=x²+y².
∂u/∂x=-y, ∂u/∂y=-x, ∂u/∂z=1
∂v/∂x=2x, ∂v/∂y=2y
∂f/∂x=∂f/∂u×∂u/∂x=-y∂f/∂u,
∂f/∂y=∂f/∂u×∂u/∂y=-x∂f/∂u,
∂f/∂z=∂f/∂u×∂u/∂z=∂f/∂u
and
∂f/∂x=2x∂f/∂v, ∂f/∂y=2y∂f/∂v