See [this note][1] describing the algebraic tangent space of a $C^k$ manifold. On $\mathbb{R}$, we can therefore define a derivation at $p$ which maps $|x-p|^{k+\frac{1}{2}}$ to $1$ and every other basis element to $0$.




  [1]: https://www.ams.org/journals/bull/1973-79-04/S0002-9904-1973-13293-8/S0002-9904-1973-13293-8.pdf