Given an L-infinity algebra, we prove a statement about the simultaneous deformation of the L-infinity algebra structure and of its Maurer-Cartan elements. The main tool is Ted Voronov's derived bracket construction. An instance is the following: if F: g--->h is a Lie algebra morphism, the simultaneous deformation of the Lie algebras structures and of the map F to a new morphism are governed by a cubic equation; our statement asserts that there exists an L-infinity algebra whose Maurer-Cartan equation is exactly this cubic equation. We will discuss other examples as well.