I'm reading a proof and I'm struggling with basic calculus here. Given the equation, $F[x, F(y, z)] = F[F(x, y), z] $ set $u = F(x, y)$ and $v = F(y, z)$ so you have
$F(x, v) = F(u, z)$
Now differentiate with respect to $x$ and $y$ leads to the following
- $F_{x}(x,v) = F_{x}(u,z)F_{x}(x,y)$
- $F_{y}(x,v)F_{x}(y,z) = F_{x}(u,z)F_{y}(x,y)$
I think, according to the chain rule, the derivative of $F[x, F(y, z)]$ or $F[x, u]$ with respect to x should equal $F_{x}(u,z)F_{x}(x,y)$. So number 1 makes sense. But when you differentiate that with respect to $y$, I don't understand how you get 2.