  This paper is a sequel to "t-structures and twisted complexes on derived
injectives" by the same authors. We develop the foundations of the
infinitesimal derived deformation theory of pretriangulated dg-categories
endowed with t-structures. This generalizes the deformation theory of abelian
categories developed by the last two authors. We show how deformations of
dg-categories of derived injectives yield derived deformations of the
associated t-structures.
