Suppose the variables $x, y, z$ are related by $z=f(x,y)$ and $y=g(x,z)$. Show that $$\frac{\partial f}{\partial y}\frac{\partial g}{\partial z}=1$$
Intuitively this makes sense. By abusing notation, one imagines switching the partials of $y$ and $z$ to obtain $$\frac{\partial f}{\partial z}\frac{\partial g}{\partial y}=1$$
But as I understand, this is not actually allowed. If it is not, how else could I prove this?
Fix $x$ and apply the chain rule to the RHS: $z=f(x,g(x,z))\Rightarrow 1=\frac{\partial f}{\partial y}\frac{\partial g}{\partial z}$.