  Let w: Map(X,Y;f) -> Y denote a general evaluation fibration. Working in the
setting of rational homotopy theory via differential graded Lie algebras, we
identify the long exact sequence induced on rational homotopy groups by w in
terms of (generalized) derivation spaces and adjoint maps. As a consequence, we
obtain a unified description of the rational homotopy theory of function
spaces, at the level of rational homotopy groups, in terms of derivations of
Quillen models and adjoints. In particular, as a natural extension of a result
of Tanre, we identify the rationalization of the evaluation subgroups of a map
f: X -> Y in this setting. As applications, we consider a generalization of a
question of Gottlieb, within the context of rational homotopy theory. We also
identify the rationalization of the G-sequence of f and make explicit
computations of the homology of this sequence. In a separate result of
independent interest, we give an explicit Quillen minimal model of a product
AxX, in the case in which A is a rational co-H-space.
