It is known that a p-adic family of modular forms does not necessarily specialize into a classical modular form at weight one, unlike the modular forms of weight 2 or higher. We will explain how this obstruction to classicality leads to a "derived" action on modular forms of weight one, which can be understood as the so-called derived Hecke operator at p. We will also investigate the role of the derived action in the study of p-adic periods of the adjoint of the weight one modular forms.