In the William Boothby book (An introduction to differentiable manifolds and Riemannian geometry) there is a definition of infinitesimal generators that using definition below, it can be proved that $X_p$ is a vector at $p$.
Definition: We define that Tangent Space $T_p(M)$ to $M$ at $P$ to be set of all mappings $X_P:C^\infty(p)\rightarrow \mathbb{R}$ satisfying for all $\alpha, \beta \in \mathbb{R}$ and $f,g \in C^\infty(p)$ the two conditions
(i) $X_p(\alpha f+ \beta g)=\alpha (X_p f)+\beta(X_p g)$ (linearity)
(ii) $X_p(fg)=(X_p f)g(p)+f(p)(X_p g)$ (Leibniz rule)
with the vector space operations in $T_p(M)$ defined by $$(X_p+Y_p)f=X_pf+Y_pf,$$ $$(\alpha X_p)f=\alpha(X_pf).$$A tangent vector to $M$ at $p$ is any $X_p \in T_p(M)$.
Now, what is the infinitesimal generator?
Suppose that $\theta:\mathbb{R} \times M \rightarrow M$ is any $C^\infty$-action. Then it defines on $M$ a $C^\infty$-vector field $X$, which we shall call the infinitesimal generator of $\theta$, according to the following prescription: For each $p \in M$ we define $X_p:C^\infty(p) \rightarrow \mathbb{R}$ by $$X_pf=\lim_{\Delta t\to 0} \frac{1}{\Delta t} [f(\theta_{\Delta t} (p))-f(p)].$$
Now, the book says that using the first definition, it is obvious that $X_p$ is a vector. Actually, the first condition is easy to check and I have done that! But, When I try to prove that it satisfies the second one, IT SUCKS!!!
So, any help is definitely appreciated!