# What is the formula for Cross partials after change of variable?

Suppose we have a function $$f(x,y) = 2x^2+2y^2$$

Next we change the variables with $$u(x,y)=2x^2+2y^2$$ so we have $$f(x,y)=g(u(x,y))=u(x,y)$$

Since these functions are equal everywhere, we can write $$f(x,y)=g(u(x,y))$$ and take partials of both sides.

Question: What is the formula for $$\frac{\partial}{\partial y}\frac{\partial}{\partial x} g(u(x,y))$$ (in general, not for the specific above example)

For example, we can say that $$\tag{1} \frac{\partial}{\partial x} f(x,y) = \frac{\partial}{\partial x} g(u(x,y)) = \frac{\partial g }{\partial u} \frac{\partial u }{\partial x}$$

But I want such a a relationship for cross partials.

(note, both the LHS and RHS of (1) both are $$4x$$ for the example I have given)

Note: I am using $$\frac{\partial}{\partial y}\frac{\partial}{\partial x}$$ to mean first take the partial w.r.t to $$x$$ and then the partial of that partial w.r.t $$y$$. I may have the order backwards; I always forget what is the correct order to write the partials in.

Sincce $$f_x=g^\prime(u)u_x$$, $$f_{xy}=g^{\prime\prime}(u)u_xu_y+g^\prime(u)u_{xy}$$.