Regular distribution on manifold

Question

Let $M$ be a compact $d$-dimensional Riemannian manifold with volume form $dV$. By analogy with the Euclidean case, call a distribution $\mu$ on $M$ (that is, a continuous linear functional on $C^{\infty}(M)$) regular if there exists $f\in L^1(M)$ such that \begin{align*} \langle \mu,\varphi\rangle = \int_M \varphi(x)f(x)dV \end{align*} for all $\varphi\in C^{\infty}(M)$. Is it true that if there exists a constant $C$ such that \begin{align*} \lvert\langle \mu, \partial_i\varphi(x)\rangle\rvert \leq C\lvert\varphi\rvert_\infty \end{align*} for all $i=1,\dots,d$ and all $\varphi\in C^{\infty}(M)$, then $\mu$ is regular?