$C^1$-Functions as the domain closure of differentiation the polynomials

Question

Let $\mathbb{R}[x]$ denote the polynoimals in one variable considered as a subalgebra of the continuous functions from $\mathbb{R}$ to $\mathbb{R}$. The operator $\frac{d}{dx}$ is a closable operator acting on $\mathbb{R}[x]$. Is the domain of its closure the $C^1$-functions, i.e. once differeniable functions on $\mathbb{R}$, or is it something more subtle?