Multiplication of a distribution by locally integrable function

Question

If $f=e^{-|x|}$ then how is it true that $f\delta'[\varphi]=-f'\delta[\varphi]-f\delta[\varphi']$ for $\varphi\in\mathcal{D}(\Omega)$?
We know that for $f\in C^{\infty}$ above estimate true but $f$ given here are not smooth. Suppose we only know that $f\in L_{loc}^1$ or $f$ is Lipschitz continuous. Then is there any way to hold the above equality?
Let if we define in the way that $f\delta_{x_0}[\varphi]=f(x_0)\varphi(x_0)$ and others terms in the same ways.
Then if we mollify $f$ then for $f_{\epsilon}\in C^{\infty}$ above result holds but the problem is that $f_{\epsilon}\to f$ pointwise a.e not for all $x_0$ as $f\in L_{loc}^1$.