Let $f: U \subseteq \mathbb{R}^2 \to \mathbb{R}^2$ where $U$ is an open subset of $\mathbb{R}^2$ be a $C^\infty$ map and let $g: U \to \mathbb{R}$ be a $C^\infty$ map with $g(u) \neq 0$ for all $u \in U$. Is it true that $f/g: U \to \mathbb{R}^2: u \mapsto f(u)/g(u)$ is a $C^\infty$-map?
A simple yes or no suffices.
The answer is yes. You can prove this using a common criterion from calculus: if $f$ and $g$ are continuously differentiable on $U$ and $g$ is nonvanishing on $U$, then $$ \frac{d}{dx}\bigg(\frac{f}{g}\bigg)=\frac{f'g-fg'}{g^2}.$$ Now if you know that a sum and product of smooth functions is smooth, you can apply induction to conclude that $f/g$ is arbitrarily many times differentiable, i.e. of class $\mathscr{C}^\infty$.