First, some notation and definitions.
Let $v=(a^1,\dots,a^n)\in T_p\mathbb{R}^n\cong\mathbb{R}^n$ be a tangent vector and let $f$ be a $C^\infty$ map defined in a neighbourhood of $p.$ The directional derivative of $f$ in $p$ with respect to $v$ is $$\partial_v f(p)=\sum_{i=1}^n a^i\frac{\partial f}{\partial x^i}(p).$$ Consider $C^\infty_p=\lbrace f:U\rightarrow\mathbb{R} | f \text{ is a } C^\infty\text{map defined in a neighbourhood } U \text{ of } p\rbrace$. Then the directional derivative at $p$ is an operator $$\partial_v|_p: C^\infty_p\longrightarrow\mathbb{R}$$ $$f\longmapsto \partial_vf(p)$$ that satisfies the following properties: $$\partial_v|_p(f+g)=\partial_v|_pf+\partial_v|_pg;$$ $$ \partial_v|_p(constant)=0; $$ $$ \partial_v|_p(fg)=g(p)\partial_v|_pf+f(p)\partial_v|_pg .$$
Operators $D:C^\infty_p\rightarrow\mathbb{R}$ that satisfy these three properties are called derivations centered at $p$.
Let $Der_p$ be the set of such operators. Notice that $Der_p$ is a vector space. We have seen that to each vector $v\in T_p\mathbb{R}^n$ is associated the derivation $\partial_v|_p=\sum_{i=1}^n a^i\frac{\partial f}{\partial x^i}|_p\in Der_p.$ In this way, we obtain a homomorphism between vector spaces $$T_p\mathbb{R}^n\longrightarrow Der_p$$ $$v\longmapsto \partial_v|_p$$
Question: how to formally show that the above is a homomorphism?