Chain rule for differential of f(x,y,g)f(x,y,g)

I’m given a function F=f(x,y,g(x,y))F=f(x,y,g(x,y)) and I want to find (∂∂xnF)(\frac{\partial}{\partial x_n} F) in terms of partial derivatives of ff and gg.
I wrote h=(x,y)→(x,y,g(x,y))h=(x,y)\rightarrow (x,y,g(x,y)), so F=f(h(x,y))F=f(h(x,y)), then tried directly applying chain rule to get ∂∂xF=∂∂hf∂∂xh\frac{\partial}{\partial x} F=\frac{\partial}{\partial h} f\frac{\partial}{\partial x} h, but then I have no idea what to do with the ff partial since it’s a partial with respect to something not defined in the problem, much less x or y.

How would I do this kind of differentiation?

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The redefinition of the argument of F is unnecessary, you can just apply the chain rule directly, for example ∂F∂x=∂f∂xdxdx+∂f∂ydydx+∂f∂g∂g∂x=∂f∂x+∂f∂g∂g∂x\frac{\partial F}{\partial x} = \frac{\partial f}{ \partial x} \frac{d x}{d x} + \frac{\partial f}{ \partial y} \frac{d y}{d x} + \frac{\partial f}{ \partial g} \frac{\partial g}{\partial x} = \frac{\partial f}{ \partial x} + \frac{\partial f}{ \partial g} \frac{\partial g}{\partial x}
– Triatticus
Oct 21 at 2:58

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