The notion of a derived A-infinity algebra arose in the work of Sagave as a natural generalisation of the classical A-infinity algebra, relevant to the case where one works over a commutative ring rather than a field. We develop some of the basic operadic theory of derived A-infinity algebras, building on work of Livernet-Roitzheim-Whitehouse. In particular, we study the coalgebras over the Koszul dual cooperad of the operad dAs, and provide a simple description of these. We study representations of derived A-infinity algebras and explain how these are a two-sided version of Sagave's modules over derived A-infinity algebras. We also give a new explicit example of a derived A-infinity algebra.