Correspondence: Derivations and vector fields

Question

Let $D$ be a derivation on the $\mathbb{R}$-algebra $C^\infty(\mathbb{R})$ of smooth functions on $\mathbb{R}$. Let $f\in C^\infty(\mathbb{R})$. By the Hadamard lemma we find a smooth function $g\in C^\infty(\mathbb{R})$ such that $f(x)=f(0)+x\cdot g(x)$ for all $x\in \mathbb{R}$. I was able to show that then the following identity holds: $$D(f)=D(\operatorname{id}_\mathbb{R})\cdot g + \operatorname{id}_\mathbb{R} \cdot D(g).$$ In particular, we have $$D(f)(0)=D(\operatorname{id}_\mathbb{R})(0)\cdot g(0).$$ I am struggling to show that, more generally, we have $$D(f)(x)=D(\operatorname{id}_\mathbb{R})(x)\cdot g(x)$$ for all $x\in \mathbb{R}$. Any hints?