I have a problem with the last line of the following proof that shows that tangent space $T_p(\mathbb{R}^n)$ is isomorphic to the space of derivations at point $p$, $\mathcal{D}_p(\mathbb{R}^n)$:
I fail to understand the last line which shows surjectivity. How does it follow from $D = D_v$, with $ v=\langle Dx^1, Dx^2, \ldots, Dx^n\rangle$.
Edit: $D_v$ is the directional derivative and derivations are defined in terms of their action on products of $C^\infty$ functions $f, g$. $$ D(fg)(p) = (Df)g(p)+(Dg)f(p) $$