Suppose $\psi(x,y)=\phi(x+y,xy)$ find $\frac{\partial^2 \psi}{\partial y \partial x}$
My attempt: I think this must be an application of the chain rule. Do first I need to take the derivative with respect to y, and then follow that up with a derivative of x? I am a little confused about how to take a derivative of a function of x and y equivalent to another function of x and y. I tried looking for some examples similar to this one but was unsuccessful.
Hint
Write $u(x,y)=x+y$ and $v(x,y)=xy.$ Then, we get the partial with respect to $x$ of $\psi(x,y)=\phi(u(x,y),v(x,y)).$ We have
$$\frac{\partial \psi}{\partial x}=\frac{\partial \phi}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial \phi}{\partial v}\frac{\partial v}{\partial x}.$$
(Note that $\frac{\partial u}{\partial x}=1$ and $\frac{\partial v}{\partial x}=y.$)
Can you continue from here? That is, can you get $\frac{\partial}{\partial y}\frac{\partial \psi}{\partial x}?$