  We provide a semiorthogonal decomposition for the derived category of
fibrations of quintic del Pezzo surfaces with rational Gorenstein
singularities. There are three components, two of which are equivalent to the
derived categories of the base and the remaining non-trivial component is
equivalent to the derived category of a flat and finite of degree 5 scheme over
the base. We introduce two methods for the construction of the decomposition.
One is the moduli space approach following the work of Kuznetsov on the sextic
del Pezzo fibrations and the components are given by the derived categories of
fine relative moduli spaces. The other approach is that one can realize the
fibration as a linear section of a Grassmannian bundle and apply Homological
Projective Duality.
