  We consider the quotient of an exact or one-sided exact category
$\mathcal{E}$ by a so-called percolating subcategory $\mathcal{A}$. For exact
categories, such a quotient is constructed in two steps. Firstly, one localizes
$\mathcal{E}$ at a suitable class $S_\mathcal{A} \subseteq
\operatorname{Mor}(\mathcal{E})$ of morphisms. The localization
$\mathcal{E}[S_\mathcal{A}^{-1}]$ need not be an exact category, but will be a
one-sided exact category. Secondly, one constructs the exact hull
$\mathcal{E}{/\mkern-6mu/} \mathcal{A}$ of $\mathcal{E}[S_\mathcal{A}^{-1}]$
and shows that this satisfies the 2-universal property of a quotient amongst
exact categories.
  In this paper, we show that this quotient $\mathcal{E} \to \mathcal{E}
{/\mkern-6mu/} \mathcal{A}$ induces a Verdier localization
$\mathbf{D}^b(\mathcal{E}) \to \mathbf{D}^b(\mathcal{E} {/\mkern-6mu/}
\mathcal{A})$ of bounded derived categories. Specifically, (i) we study the
derived category of a one-sided exact category, (ii) we show that the
localization $\mathcal{E} \to \mathcal{E}[S_\mathcal{A}^{-1}]$ induces a
Verdier quotient $\mathbf{D}^b(\mathcal{E}) \to
\mathbf{D}^b(\mathcal{E}[S^{-1}_\mathcal{A}])$, and (iii) we show that the
natural embedding of a one-sided exact category $\mathcal{F}$ into its exact
hull $\overline{\mathcal{F}}$ lifts to a derived equivalence
$\mathbf{D}^b(\mathcal{F}) \to \mathbf{D}^b(\overline{\mathcal{F}})$.
  We furthermore show that the Verdier localization is compatible with several
enhancements of the bounded derived category, so that the above Verdier
localization can be used in the study of localizing invariants, such as
non-connective $K$-theory.
