The definition of $D_v$ (maybe this is directional derivative) and a property(Leibnitz rule) are written as follows, but I wonder this is correct.
Let $X$ be manifold, $U\subset X$ be open, $p\in U.$
For $v=[\ \ell \ ]\in T_p X$, define $D_v : C^\infty (U,\mathbb R) \to \mathbb R$ as $f\mapsto \dfrac{d}{dt}\Bigg|_{t=0} \dfrac{d(f\circ \ell)}{dt}(t)$.
Then we can see $D_v$ satisfies Leibnitz rule ; $D_v(fg)=f(p)D_v(g)+g(p)D_v (f)$.
I think $D_v$ is defined as directional derivative. If so, I think $D_v$ should be defined as $f\mapsto \dfrac{d}{dt}\Bigg|_{t=0}(f\circ \ell)(t) \left(=(f\circ \ell)'(0)\right)$.
If I use the definition $f\mapsto \dfrac{d}{dt}\Bigg|_{t=0}(f\circ \ell)(t) \left(=(f\circ \ell)'(0)\right)$, I can see the Leibnitz rule by simple calculation, but if I use the definition $f\mapsto \dfrac{d}{dt}\Bigg|_{t=0} \dfrac{d(f\circ \ell)}{dt}(t)$, I cannot see the Leibnitz rule, so I think this definition $f\mapsto \dfrac{d}{dt}\Bigg|_{t=0} \dfrac{d(f\circ \ell)}{dt}(t)$ is wrong.
Is this mistake of author ?
It is wrong, but certainly a typo. It would be correct to write $$D_v(f) = \dfrac{d(f\circ \ell)}{dt}(t)\Bigg|_{t=0}$$ or $$D_v(f) = \dfrac{d}{dt}\Bigg|_{t=0} (f\circ \ell)(t) .$$