In a manifold it's easy to motivate the definition of vectors as equivalence classes of curves. On the other hand the definition as derivations is harder to motivate. I know how to show that the space obtained with derivations is isomorphic to the space obtained with curves, so I know that defining with derivations works.
My doubt it is: how would I motivate the usage of derivations? This question is, how can it be intuitively visible that linearity and Leibniz rule alone are sufficient for assuring that the operator is a directional derivative?
I've seem some discussion about it on the "Applied Differential Geometry" book from William Burke where he tries to motivate this saying that a differential operator measures terms in the Taylor series and that Leibniz rule assures the operator is sensible only to first order terms. But I didn't get this idea yet.