It is shown that the invariance of the S-matrix under the addition of a 4-divergence to the Lagrangian density of a system of interacting fields allows one to derive various equivalence theorems in a very simple manner. A simple illustration of this method is followed by a discussion of a class of derivative couplings for which equivalence theorems can be proved. Applications of some exact equivalence theorems are then made to a renormalizability problem and to the question of the generation of vector fields from local gauge invariance. Finally, the same invariance property is employed to prove the theorem that point transformations of the fields leave the S-matrix unchanged.