# Estimate $\partial^\alpha |x|$ in $\mathbb{R}^n\setminus\{0\}$

I'm working with the derivative of a radial function $$f(r)\in C^\infty(\mathbb R),$$ so I need to calculate $$\partial^\alpha f(|x|)$$ for $$x$$ in $$\mathbb{R}^n$$ not equal to the origin. $$\alpha\in \mathbb{N}^n$$ is a multi-index. By chain rule, I have to calculate $$\partial^\beta|x|,$$ $$\forall \beta\le\alpha.$$ But I find it quite hard to give an explicit expression.

I try to estimate $$|\partial^\alpha f(|x|)|$$ then, which means I have to estimate $$|\partial^\alpha|x||$$, $$\forall \alpha\in \mathbb{N}^n.$$ I guess we have $$|\partial^\alpha|x||\le 1$$ but fail to prove it. I tried induction but it's too complex. I have to consider many cases of $$\alpha$$ be like.

Is there an elegant way to derive or at least estimate $$\partial^\alpha|x|$$?

There is a good way to induct. For $$k\in\mathbb R$$, let's call a function $$f:\mathbb R^n\setminus \{0\} \to \mathbb R$$ $$k$$-homogeneous if $$f(\lambda x) =\lambda^k f(x)$$ for all $$x\in \mathbb R^n\setminus \{0\}$$ and $$\lambda > 0$$. For example, $$|x|$$ is $$1$$-homogeneous. $$0$$ happens to be $$42$$-homogenous. With this concept in mind, its not hard to prove
Prop. Let $$f:\mathbb R^n\setminus \{0\} \to \mathbb R$$ be smooth and $$k$$-homogeneous. Then any partial $$\partial_i f$$ is $$(k-1)$$-homogeneous.
Proof. \begin{align} (\partial_i f)(\lambda x) &= \lim_{h\to 0}\frac{f(\lambda x+he_i)-f(\lambda x)}h \\&=\lambda^k\lim_{h\to 0}\frac{f( x+he_i/\lambda)-f( x)}h \\&=\lambda^{k-1}\lim_{h\to 0}\frac{f( x+(h/\lambda)e_i)-f( x)}{h/\lambda} \\&=\lambda^{k-1}\lim_{t\to 0}\frac{f( x+te_i)-f( x)}t \\&= \lambda^{k-1} \partial_i f(x). \text{QED} \end{align}
Corollary. $$|\partial^\beta|x|| \le C_{\beta} |x|^{1-|\beta|}.$$ ($$C_{\beta}$$ is the maximum of the smooth function $$\partial^\beta|x|$$ on the unit sphere.)
• Thanks a lot. It also solve my another question: What if taking $\ell^p$ norm of $x$ instead $p=2$? Now taking $\ell^p$ norm is also $1$-homogeneous so the answer is same. Nov 11, 2021 at 15:09