  This paper investigates the transformation of determinants of pairs of
Fredholm operators with trace class commutators. We study the extent to which
the functional calculus commutes, modulo operator ideals, with projections in a
finitely summable Fredholm module. As an application, we recover in particular
some results of R. Carey and J. Pincus on determinants and Tate tame symbols.
Additionally, we obtain variational formulas for joint torsion.
