Let $f: \mathbb{R}^3 \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}$ 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 $f$.