let $M$ be a smooth manifold. let $x \in M$. i want to define the tangent space $T_xM$ as the space of all derivations over smooth functions at $x$ (with appropriate addition and scalar multiplication defined on the derivations).
I am a bit confused because wikipedia states that a derivation as $D: C^\infty(M) \to \mathbb{R}$, so it sends a smooth function $f: M \to \mathbb{R}$ to a real number.
Is that right? I would have thought that it should send the smooth function evaluated at the point $x$ to a real number, so $f(x = x') \mapsto Df(x) \lvert_{x = x'}$