Let $F : \mathbb{R}^2 \to \mathbb{R}$ be defined by the equation $F(x, y) = f(x, y, g(x, y))$. (a) Find $DF$ in terms of the partials of $f$ and $g$.

Question

Let $f : \mathbb{R}^3 \to \mathbb{R}$ and $g : \mathbb{R}^2 \to \mathbb{R}$ be differentiable. Let $F : \mathbb{R}^2 \to \mathbb{R}$ be defined by the equation $F(x, y) = f(x, y, g(x, y))$.

(a) Find $DF$ in terms of the partials of $f$ and $g$.

(b) If $F(x , y) = 0$ for all $(x, y)$, find $D_1g$ and $D_2g$ in terms of the partials of $f$.

I have already seen a solution to this exercise and it is as follows:

But I do not understand why $\begin{bmatrix} D_1F(p) & D_2F(p) \end{bmatrix}=\begin{bmatrix} D_1f(q)& D_2f(q) &D_3f(q) \end{bmatrix}\begin{bmatrix} 1 & 0\\ 0 & 1\\ D_1g(p) &D_2g(p) \end{bmatrix}$? Could anyone help me, please? Thank you very much.