This is joint work with Greg Lupton, Cleveland State Univ. Let Map(X, Y; f) denote the space of functions from X to Y homotopic to f. Let w : Map(X, Y;f) -> Y denote the evaluation map. We identify w on rational homotopy groups in terms of (generalized) derivation spaces of differential graded Lie algebras and adjoint maps. As a consequence, we express the rationalization of a question of Gottlieb on the difference between evaluation subgroups and Whitehead centralizers in terms of a particular commutative diagram.