Distributional and classical derivative

Question

Let $\Omega$ be an open subset of $ \mathbb{R}^n$, and $f\in L^1_{loc}(\Omega)$. Given a multiindex $\alpha$, suppose that the $\alpha$-distributional derivative $D^\alpha T_f$ is a distribution arising from a function $g \in L^1_{loc}(\Omega)$. I’m wondering if it implies that $g$ is equal to the classical derivative $D^\alpha f$ almost everywhere (provided that $D^\alpha f$ almost every exists almost everywhere).