I'm revisiting the definition for tangent spaces in Lee's Introduction to Smooth Manifolds and I'm trying to convince myself why we might define tangent vectors as derivations at a point $p\in M$:
Let $M$ be a smooth manifold, and let $p\in M$. A linear map $v:C^\infty(M)\to \mathbb{R}$ is called a derivation at $p$ if \begin{align*} v(fg) = f(p)vg + g(p)vf \end{align*} for all $f,g\in C^\infty(M)$.
So far, I know that if $M=\mathbb{R}^n$, then each derivation can be given as a directional derivative in some direction in $\mathbb{R}^n$. After reading the parts on the differential and its computation in coordinates, I'm still wondering why we would be interested in defining a tangent vector as a map that acts on functions on the manifold and the benefits from acting on smooth functions. The main reason that I can think of is that the collection of derivations at a point forms a vector space, which is we what want for a tangent space.
I have also looked at the approach of defining tangent vectors with equivalence classes of curves, but it seems that there's also an action on $f\in C^\infty(M)$ going on; we call curves $\gamma:J\to M$ the tangent vectors, and they have a directional-derivative-like operators that act on $f\in C^\infty(M)$ by \begin{align*} \left.\frac{d}{dt}(f\circ \gamma)(t)\right|_{t=0}. \end{align*} This seems really similar to how a vector in $\mathbb{R}^n$ defines its own directional derivative, but again, I'm not sure why the action on $f\in C^\infty(M)$ would be useful/significant.
