We elaborate on a recently proposed extension of the Gerasimov-Drell-Hearn (GDH) sum rule which is achieved by taking derivatives with respect to the anomalous magnetic moment. The new sum rule features a linear relation between the anomalous magnetic moment and the dispersion integral over a cross section quantity. We find some analogy of the linearized form of the GDH sum rule with the “sideways dispersion relations.” As an example, we apply the linear sum rule to reproduce the famous Schwinger’s correction to the magnetic moment in QED from a tree-level cross section calculation and outline the procedure for computing the two-loop correction from a one-loop cross section calculation. The polarizabilities of the electron in QED are considered as well by using the other forward-Compton-scattering sum rules. We also employ the sum rules to study the magnetic moment and polarizabilities of the nucleon in a relativistic chiral effective field theory (EFT) framework. In particular we investigate the chiral extrapolation of these quantities.