Tangent space to a manifold - First Order Approximation to the Manifold 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.
 A: In the case of a function $f:\Bbb R^n\to\Bbb R,$ the first order approximation to $f$ at a point $x$ is a linear function, say $Df,$ which agrees with $f$ at $x,$ and closely approximates nearby values of $f.$ Geometrically, it turns out that the graph of $Df$ is a hyperplane locally tangent to the graph of $f.$
In the case of manifolds, a priori we may not have a local embedding, as for graphs, and more importantly, the manifold itself need not be defined as the graph of a particular function. So strictly speaking, there is no first order approximation of some function lurking around. But, it still makes sense to consider a vector space which locally best approximates the manifold. The way this is done is via derivatives and derivations, which ultimately carry much more structure than a simple vector space would, letting us do calculus locally on manifolds, but intuitively one can just think of the tangent space as a local approximation of the manifold.
A: In principle the tangent space T_pM is an abstract thing. But any manifold can be embedded in some R^n (Whitney theorem), and there T_pM is a true linear subspace (up to a translation) and the tangent vector to a curve in M is a true vector and so on... so you can do the same reasoning that you do in analysis
