Let $M = \mathbb{R}$ and check that the following map is a derivation at $p \in \mathbb{R}$:
$$
D(f) = 
\begin{cases}
\lim\limits_{x \to p} \dfrac{f(x) - f(p)}{\;|x-p|^{r + \frac{1}{2}}} &\text{if the limit exists} \\[1ex]
0 &\text{otherwise} 
\end{cases}.
$$