Partial derivative of an iterated function

Question

I'm trying to find the form of the partial derivative for the following function:

$F(F(x_{1},x_{2}),x_{2})$

I tried using the rules of composition:

$\frac{\partial F}{\partial x_{2}}=\frac{\partial F}{\partial F}\frac{\partial F}{\partial x_{2}}+\frac{\partial F}{\partial x_{2}}\frac{\partial x_{2}}{\partial x_{2}}$

And this was confusing so I changed the name of the external function to G:

$\frac{\partial G}{\partial x_{2}}=\frac{\partial G}{\partial F}\frac{\partial F}{\partial x_{2}}+\frac{\partial G}{\partial x_{2}}\frac{\partial x_{2}}{\partial x_{2}}$

And here I cannot find a way out. Because the result is $\frac{\partial G}{\partial x_{2}}$ equal to itself plus another component. Can someone help me to understand my mistake please?

Answer 1

Yes, this is what happens when one reuses a letter in writing the chain rule.

Let's write $G(x_1,x_2) = F(F(x_1,x_2),x_2)$. Then $$\frac{\partial G}{\partial x_2}(x_1,x_2) = \frac{\partial F}{\partial x_1}(F(x_1,x_2),x_2)\frac{\partial F}{\partial x_2}(x_1,x_2) + \frac{\partial F}{\partial x_2}(x_1,x_2).$$