Abstract: The Gabriel-Popescu Theorem states essentially that every Grothendieck category is (up to equivalence) of the form (R,σ)-mod, i.e., the quotient category of some left module category R-mod, by some Serre subcategory Tσ associated to an idempotent kernel functor σ. It follows that every Grothendieck category is a Giraud suncategory of a left module category. In this paper, we study the relationship between derived functors in a Grothendieck category C and a Giraud subcategory D. With C=R-mod, the foregoing thus yields methods to study derived functors in arbitrary Grothendieck categories from knowledge about derived functors in R-mod.