Let $p(x),q(x)\in\mathbb{R}[x]$ be polynomials, and define

Then, for $x\neq 0$,

Computing (e.g. by applying L’Hôpital’s rule), we see that $g^{{\prime}}(0)=\lim_{{x\to 0}}g^{{\prime}}(x)=0$.

Define $p_{0}(x)=q_{0}(x)=1$. Applying the above inductively, we see that we may write $f^{{(n)}}(x)=\frac{p_{n}(x)}{q_{n}(x)}f(x)$. So $f^{{(n)}}(0)=0$, as required.