We show how higher (simplicial or differential graded) techniques are naturally required when studying infinitesimal deformation theory. We then outline the strict relationship between derived moduli spaces and the combination of classical moduli spaces plus infinitesimal derived/extended/higher deformation functions, as the former induce the latter, but the latter carry (in an appropriate sense) all the information of the former. In the process, we consider explicit examples and highlight the role of differential graded Lie algebras and their generalisations.