Change varibles wrt to partial derivative

So we have a function in terms of x and y and we want to change to some other co-ordinate system say u and v. f(x,y) goes to h(u,v). So with respect to partial derivatives I can express
df/dx = dh/du * du/dx + dh/dv * dv/dx where d is of course the "curly d".

My question is what is the formula for d^2f/dx^2 or d^2f/dxdy in this new system of co-ordinates?

You should be able to work it out yourself from what you have already, really.

∂f/∂x = ∂h/∂u ∂u/∂x + ∂h/∂v ∂v/∂x
So ∂^2f/∂x^2 = (∂^2h/∂u^2 ∂u/∂x + ∂^2h/∂u∂v ∂v/∂x) ∂u/∂x + (∂h/∂u) ∂^2u/∂x^2 + (∂^2h/∂u∂v ∂u/∂x + ∂^2h/∂v^2 ∂v/∂x) ∂v/∂x + (∂h/∂v) ∂^2v/∂x^2
= (∂^2h/∂u^2) (∂u/∂x)^2 + 2(∂^2h/∂u∂v) (∂u/∂x ∂v/∂x) + (∂^2h/∂v^2) (∂v/∂x)^2 + (∂h/∂u) (∂^2u/∂x^2) + (∂h/∂v) (∂^2v/∂x^2)

and ∂^2f/∂x∂y = (∂^2h/∂u^2 ∂u/∂y + ∂^2h/∂u∂v ∂v/∂y) ∂u/∂x + (∂h/∂u) ∂^2u/∂x∂y + (∂^2h/∂u∂v ∂u/∂y + ∂^2h/∂v^2 ∂v/∂y) ∂v/∂x + (∂h/∂v) ∂^2v/∂x∂y.
= (∂h^2/∂u^2) ∂u/∂x ∂u/∂y + (∂^2h/∂u∂v) (∂u/∂x ∂v/∂y + ∂u/∂y ∂v/∂x) + (∂^2h/∂v^2) ∂v/∂x ∂v/∂y + (∂h/∂u) (∂^2u/∂x∂y) + (∂h/∂v) (∂^2v/∂x∂y).