I've a doubt about the tangent space to a manifold. Let $M$ be a $n$-manifold and let $p\in M$, I've heard that the tangent space $T_pM$ at $p$ is the first order approximation of $M$ near $p$ in the same way that the tangent hyperplane to the graph of a function $f : \mathbb{R}^n \to \mathbb{R}$ is the first order approximation to the graph of $f$.
This is really intuitive, but how do I show that ? I mean, I'm using the definition of tangent space with derivations, how do I show that that abstract set associated with each point of the manifold gives the first order approximation to the manifold ? Is this fact already built in into the definition somehow or we should prove it ? If we should prove it, can someone give a hint ? I don't want the full proof, just a hint to begin the proof.
Thanks in advance for your aid, and sorry if this question is too trivial.