  In this work, we use semigroup integral to evaluate zeta-function regularized
determinants. This is especially powerful for non--positive operators such as
the Dirac operator. In order to understand fully the quantum effective action
one should know not only the potential term but also the leading kinetic term.
In this purpose we use the Weyl type of symbol calculus to evaluate the
determinant as a derivative expansion. The technique is applied both to a
spin--0 bosonic operator and to the Dirac operator coupled to a scalar field.
