Certain identities on derivatives of radial homogeneous and logarithmic functions

Let $k$ be a natural number and $s$ be real. In the 1-dimensional case, the $k$-th order derivatives of the functions $lvert xrvert^s$ and $log lvert xrvert$ are multiples of $lvert xrvert^{s-k}$ and $lvert xrvert^{-k}$, respectively. In the present paper, we generalize this fact to higher dimensions by introducing a suitable norm of the derivatives, and give the exact values of the multiples.