  We provide a prorepresenting object for the noncommutative derived
deformation problem of deforming a module $X$ over a differential graded
algebra. Roughly, we show that the corresponding deformation functor is
homotopy prorepresented by the dual bar construction on the derived
endomorphism algebra of $X$. We specialise to the case when $X$ is
one-dimensional over the base field, and introduce the notion of framed
deformations, which rigidify the problem slightly and allow us to obtain
derived analogues of the results of Ed Segal's thesis. Our main technical tool
is Koszul duality, following Pridham and Lurie's interpretation of derived
deformation theory. Along the way we prove that a large class of dgas are
quasi-isomorphic to their Koszul double dual, which we interpret as a derived
completion functor; this improves a theorem of Lu-Palmieri-Wu-Zhang. We also
adapt our results to the setting of multi-pointed deformation theory, and
furthermore give an analysis of universal prodeformations. As an application,
we give a deformation-theoretic interpretation to Braun-Chuang-Lazarev's
derived quotient.
