Let $\alpha(t)$ be a curve on, say, $\mathbb R^3$. Then $\alpha'(t)_{\alpha(t)}$ is a vector(arrow) from $\alpha$ to $\alpha + \alpha'$.
For a vector field $F:\mathbb R^3 \to \mathbb R^3$,its derivative-version $F_*:T_p(\mathbb R^3)\to T_p(\mathbb R^3)$ is a function of tangent vector field such that $F_*(\alpha'(t))=\beta'(t)$ while $\beta(t)=F(\alpha(t))$.
Then What is the function that works with the second derivative? $G_* : T_p(\mathbb R^3)\to T_p(\mathbb R^3)$ such that $G_*(\alpha'')=\beta''$?
Would $F_*$ work?