  We study the divided power structures over a product of operads with
distributive law. We give a systematic method to characterise the divided power
algebras over such a product from the structures of divided power algebra
coming from each of the the factor operads. We characterise divided power
algebras with operadic derivation, as well as divided power $p$-level algebras
in characteristic $p$, and divided power Poisson algebras in characteristic 3.
