How to show the following bound:
Let $k \in (0,\infty)$ and let $ |x|>a>0$, then there exists $C>0$ sucht that for all $m\ge 1$
\begin{align} \left| \frac{d^m}{dx^m} e^{-|x|^k} \right| \le C^{m+1} |x|^{2(k-1)}e^{-|x|^k}, \forall |x|>a \end{align}
This question was raised here where it was also pointed out that the bound is almost there but not correct. The proof by induction was also suggested, but I am not sure how to do that.