A simplified proof is given of the known result that the effective potential for a given Lagrangian is equal to the sum of all connected one-particle-irreducible vacuum graphs belonging to a derivative lagrangian. It is pointed out, however, that the vacua of the original and derivative Lagrangians do not always coincide, and that the standard formulae must be modified to take account of this fact. Otherwise, for some values of the classical fields, one obtains paradoxical results, such as the complexity of the one-loop contribution. It is shown that a satisfactory modification is to replace the classical field in the standard formulae by an effective field, which is a function of the classical field determined by the potential minimum. The prescription is applied to two examples, namely, a single scalar field and a spontaneously broken SO( N) potential, and it is found that in each case the effective potential is real and finite, but may have phase transitions, i.e. discontinuities in the first derivatives.