Alternative 'differential' map on manifold

Question

Given $a, b: I \to M$, smooth curves on some manifold M, $a(0) = b(0) = p$ and $a'(0) = b'(0) = 0$. Prove that the map $D_p: \mathcal{F}(M) \to \mathbb{R}$ so that $D_p(f) = (f\circ a)''(0)-(f\circ b)''(0)$ is linear and satisfies Leibniz rule. Find the corresponding vector in coordinates.
It have a couple of similar exercises, that is why I would like to have a proof of this as an example.