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 also be understood as the so-called derived
Hecke operator at p. We will discuss the role of the derived action in
the study of p-adic periods of the adjoint of the weight one modular
forms.