Denote $|x|$ the Euclidean norm of a vector $x\in\mathbb R^N$. Also denote $D^kf$ as the vector in $\mathbb R^{N^k}$ containing all $k$-order partial derivatives of the function $f\colon\mathbb R^N\to\mathbb R$. Consider the following norm for $D^kf(x)\in\mathbb R^{N^k}$:
$$\|D^kf(x)\|=\left[\sum_{i_1,\ldots,i_k=1}^N\left(\dfrac{\partial^kf}{\partial x_{i_1}\cdots\partial x_{i_k}}(x)\right)^2\right]^{\frac12}. $$
Now let $f(x)=|x|$. My question is if the following equation holds for some $C_{N,k}\in\mathbb R$:
$$\|D^kf(x)\|=\dfrac{C_{N,k}}{|x|^{k-1}},\quad\forall x\in\mathbb R^N\backslash\{0\}\mbox{ and }k\in\mathbb N=\{1,2,\ldots\}. $$
The case $k=1,2,3$: I was able to prove for those cases because
$$\dfrac{\partial f}{\partial x_i}(x)=\dfrac{x_i}{|x|},\ \dfrac{\partial^2f}{\partial x_i\partial x_j}(x)=\dfrac{\delta_{ij}}{|x|}-\dfrac{x_ix_j}{|x|^3},$$and
$$\dfrac{\partial^3f}{\partial x_i\partial x_j\partial x_k}(x)=\dfrac{3x_ix_jx_k}{|x|^5}-\dfrac{\delta_{jk}x_i}{|x|^3}-\dfrac{\delta_{ij}x_k}{|x|^3}-\dfrac{\delta_{ik}x_j}{|x|^3}.$$
After doing the calculations, we have
$$\|D^1f(x)\|=1,\ \|D^2f(x)\|=\dfrac{\sqrt{N-1}}{|x|},\mbox{ and }\|D^3f(x)\|=\dfrac{\sqrt{3N-3}}{|x|^2}.$$
Therefore, the question is true for $k=1,2,3$ with $C_{N,1}=1$, $C_{N,2}=\sqrt{N-1}$, and $C_{N,3}=\sqrt{3N-3}$.
My try: I only have two types of ideas that seem promising. First idea: proving by induction on $k$. However, I was not able to use the induction hypothesis properly.
Second idea: for $k\geq3$, we have
$$\dfrac{\partial^k}{\partial x_{i_1}\cdots\partial x_{i_k}}\left(|x|^2\right)=0.$$
Unfortunately, I was not able to develop the left-term (using chain rule) to isolate the term $\frac{\partial^kf}{\partial x_{i_1}\cdots\partial x_{i_k}}(x)$ to calculate $\|D^kf(x)\|$.
A last attempt (less promising) is to develop by "brute force" the term $\frac{\partial^kf}{\partial x_{i_1}\cdots\partial x_{i_k}}(x)$ to obtain an expression for it which should be formally proved by induction on $k$.