The notion of relative derived category with respect to a subcategory is introduced. A triangle-equivalence, which extends a theorem of Gao and Zhang [Gorenstein derived categories, emph{J. Algebra} textbf{323} (2010) 2041-2057] to the bounded below
case, is obtained. Moreover, we interpret the relative derived functor $mathrm{Ext}_{mathcal{X}mathcal {A}}(-,-)$ as the morphisms in such derived category and give two applications.
