I'm reading a lecture notes in harmonic analysis, and I found this estimate in it:
The term $\Big|D^{\beta_1}\big(|\xi|^k\big)D^{\beta_2}\big(e^{-2\pi y|\xi|}\big)D^{\beta_3}(m(\xi))\Big|$ is dominated by a sum involving the terms of the form $|\xi|^{k-|\beta_1|}y^{|\beta_2|}e^{-2\pi y|\xi|} \left|D^{\beta_3}(m(\xi))\right|$.
$k$ is a positive integer, $y>0$ and $\beta_1,\beta_2,\beta_3$ are multi-index with $|\beta_1|+|\beta_2|+|\beta_3|=k$. $D^{\beta}$ is the (mixed) derivative. $m(\xi)$ is some function with certain properties but it is not relevant to my question.
I'm confused because we are differentiating with respect to $\xi$, not $|\xi|$, so the chain rule must be used. I don't understand why can we simple differentiate it with respect to $|\xi|$.
In fact, I could understand why $D^{\beta_1}\big(|\xi|^k\big)$ is bounded by $|\xi|^{k-|\beta_1|}$, but the $e^{-2\pi y|\xi|}$ term really confused me. Any help will be appreciated!