Assume that $f:\mathbb R^3\to \mathbb R$ and $g:\mathbb R^2 \to \mathbb R$ are two differentiable functions and $F:\mathbb R^2\to \mathbb R$ is another function with this formula:
$F(x,y):=f(x,y,g(x,y))$Write the matrix representation of $DF(x,y)$ in terms of the partial derivatives of $f$ and $g$.
Also, If $\forall (x,y)\in \mathbb R^2\quad F(x,y)=0$, Find the formula of $Dg(x,y)$ in terms of the partial derivatives of $f$.
I know some basic things about matrix representation of a differentiable function but I have no intuition what will happen in this composition and furthermore, how could we find $Dg(x,y)$ in the second case?
This is just chain rule.
\begin{align*} DF(x_0,y_0) &= D(f \circ (x,y,g(x,y)))(x_0,y_0) \\ & = D f(x_0,y_0, g(x_0,y_0)) \circ D(x,y,g(x,y)) (x_0,y_0)\end{align*}
If you want to call $G(x,y) = (x,y,g(x,y))$ then you have,
$$ DF(x_0,y_0) = Df(G(x_0,y_0))) \circ DG(x_0,y_0)$$
You know the matrix representation for each of the above differentials and so you know the composition. Also, use different coordinates for $f$, say $(u,v,w)$, then $Df(\textbf{p})$ makes sense.