Given $a\in\mathbb{R}^n$, let $T_a\mathbb{R}^n$ denote the tangent space at $a$ and $\mathcal{D}_a\mathbb{R}^n$ denote the space of derivations at $a$ on $C^{\infty}(\mathbb{R}^n)$.
We know that there is a natural isomorphism $\varphi:T_a\mathbb{R}^n \to \mathcal{D}_a\mathbb{R}^n$ by $v\mapsto D_v|_a$. In every proof I've seen of this statement, showing that $\varphi$ is surjective involves Taylor's Theorem and looks something like this.
Is there another way to show that $\varphi$ is surjective?