A major theme in ergodic Ramsey theory is proving multiple recurrence results for measure-preserving actions of semigroups. What often lies at the heart of these results is that mixing (\(\approx\) "chaotic") along a suitable filter on the semigroup amplifies itself to multiple mixing (\(\approx\) even more "chaotic") along the same filter. This amplification is usually proved using a so-called van der Corput difference lemma. Instances of this lemma for specific filters have been proven before by Furstenberg, Bergelson–McCutcheon, and others, with a somewhat different proof in each case. We define a notion of differentiation for subsets of semigroups and isolate a class of filters that respect this notion. The filters in this class (call them \(\partial\)-filters) include all those, for which the van der Corput lemma was known, and our main result is a van der Corput lemma for \(\partial\)-filters, which thus generalizes its previous instances. This is done via proving a Ramsey theorem for graphs on the semigroup.