Let $I \subseteq \mathbb{R}$ be a compact interval. We know that functions in $L^p(I)$, $(p \geq 1)$ can be $L^1$-approximated by a sequence $(\varphi_n)_{n \in \mathbb{N}}\subseteq C_0^\infty(I)$ (by that I mean $C^\infty$ with compact support).
Now, what if we want to classify the functions that can be $L^1$-approximated by derivatives of such test functions, i.e. those functions that satisfy $$ \int_I \lvert \varphi'_n -f \rvert~\mathrm{d}x \longrightarrow 0 $$ where $(\varphi_n)_{n \in \mathbb{N}}\subseteq C_0^\infty(I)$. Clearly, the functions that are derivatives of elements in $H_0^1(I)$ have this property. Can we find a broader classification?