If $f, g: S \subset \mathbb{R}^n \rightarrow \mathbb{R}$ are differentiable scalar fields, then are $fg$ and $\frac{1}{f}$ differentiable?

Question

Let $f, g: S \subset \mathbb{R}^n \rightarrow \mathbb{R}$ be scalar fields differentiable at an interior point a of $S$.

Do these results hold for scalar fields from $\mathbb{R}^n \rightarrow \mathbb{R}$ just like they do for functions from $\mathbb{R} \rightarrow \mathbb{R}$?

Is $fg$ differentiable at a? If $f(\textbf{a}) \neq 0$, is $\frac{1}{f}$ differentiable at a?