By an observation of Reid, any birational morphism between algebraic varieties can be realized through variation of geometric invariant theory quotients (VGIT). As a consequence, one way of comparing the derived categories of birational algebraic varieties is through VGIT. It turns out that this can be done even in singular cases by, in some sense, "deriving" the VGIT procedure. I will discuss how this can be done very concretely by studying VGIT for commutative differential graded algebras. This talk is about joint work with Chidambaram motivated by work with Ballard and Diemer.