Let $$ f_k (\xi)= (1+|\xi|^2)^{-k}$$ for $\xi \in R^n$ and $2k >n$. Obviously, $f_k$ is in $C^{\infty}(R^n)$. For a multi-index $\beta \in Z_{\geq 0}^n$, one can use the Leibniz rule to determine $$\partial^{\beta}(\xi^{\alpha}f_k(\xi)) = \sum_{\gamma \leq \beta} {{\beta} \choose {\gamma}} \partial^{\gamma}(\xi^{\alpha})\partial^{\beta-\gamma}(f_k(\xi))$$ for all multi-indeces $\alpha \in Z_{\geq 0}^n$ with $|a|\leq 2k-n-1$.
Here is what I am confused about: my literature only says that "all derivatives of $\xi \mapsto \xi^{\alpha}f_k(\xi)$ are linear combinations of $\xi^{\delta}\partial^{\tilde{\delta}}f_k(\xi)$ with $|\delta| + |\tilde{\delta}| \leq |\alpha|$". Using the Leibniz rule just like above, I totally don't see why that is supposed to hold. Just look at the term for $\gamma = 0$ as an example which results in a term that looks something like $\xi^{\alpha}\partial^{\beta}f_k(\xi)$ where $|\alpha|+|\beta|>|\alpha|$ for some $\beta$. There must be somehting very obvious that I am missing.