Let $\tau = (\tau_1,\ldots,\tau_n) \in \mathbf{R}^n$, $s\geq 0$, and define the function $f : \mathbf{R}^n \to \mathbf{R}$ by $$ f(\xi) = \left(\dfrac{1+ |\xi + \tau|^2}{1 + |\xi|^2}\right)^{s/2}. $$ A proof I'm reading says that for $\alpha_1,\ldots,\alpha_n$ positive integers $$ |\partial_{\xi_1}^{\alpha_1}\cdots \partial_{\xi_n}^{\alpha_n}f(\xi_1,\ldots,\xi_n)| \leq C\dfrac{(1+|\tau|)^s}{(1 + |\xi|)^{\alpha_1+\cdots+\alpha_n}}, $$ where $C$ is a constant depending on $\alpha_1,\ldots,\alpha_n,s$.
Just to see where it led, I computed a single partial and the result was already pretty nasty. There must be some clever way of obtaining such a bound, but I have no idea on how to do it. I thought that maybe homogeneity may be used, but I don't know how to rewrite $f$ in order to obtain a homogeneous function (since $f$ itself is not homogeneous). Any ideas? Thanks!