See this note describing the algebraic tangent space of a $C^k$ manifold. In particular, on $\mathbb{R}$, there exists a basis of the cotangent space $\mathfrak{m}_p\, / {\mathfrak{m}_p}^2$ containing the functions $|x-p|^{k+\alpha}$ for all $\alpha \in (0, 1)$.
We can therefore define a derivation at $p$ which maps $|x-p|^{k+\frac{1}{2}}$ to $1$ and every other basis element to $0$.