  In this paper, we study the counterpart of Grothendieck's projectivization
construction in the context of derived algebraic geometry. Our main results are
as follows: First, we define the derived projectivization of a connective
complex, study its fundamental properties such as finiteness properties and
functorial behaviors, and provide explicit descriptions of their relative
cotangent complexes, We then focus on the derived projectivizations of
complexes of perfect-amplitude contained in $[0,1]$. In this case, we prove a
generalized Serre's theorem, a derived version of Beilinson's relations, and
establish semiorthogonal decompositions for their derived categories. Finally,
we show that many moduli problems fit into the framework of derived
projectivizations, such as moduli spaces that arise in Hecke correspondences.
We apply our results to these situations.
