can a bounded ratio of two polynomials have unbounded derivative? Let $P(x_1,\cdots,x_m)$ and $Q(x_1,\cdots,x_m)$ be two polynomials and assume that $$f(x_1,\cdots,x_m)=\frac{Q(x_1,\cdots,x_m)}{P(x_1,\cdots,x_m)}$$ is bounded over some region $D\subseteq \mathbb{R}^m$.
Can $f$ have an unbounded derivative over $D$? By unbounded I mean that $|\nabla f|\rightarrow \infty$ somewhere in $D$.

tl; dr: Yes. (!!)

In Cartesian plane coordinates $(x, y)$, let
$$
P(x, y) = y,\qquad
Q(x, y) = x.
$$
Formally,
$$
f(x, y) = \frac{y}{x},\qquad
\nabla f(x, y) = \biggl(-\frac{y}{x^{2}}, \frac{1}{x}\biggr).
$$
Particularly, $|y|/x^{2} < |\nabla f(x, y)|$. It remains only to find a region where $y/x$ is bounded and $|y|/x^{2}$ is not, such as
$$
D = \{(x, y) : 0 < y < |x|^{3/2}\}.
$$
