  We study a computational method of the hyperbolic Reidemeister torsion (also
called in the literature the non-abelian Reidemeister torsion) induced by J.
Porti for complete hyperbolic three-dimensional manifolds with cusps. The
derivative of the twisted Alexander invariant for a hyperbolic knot exterior
gives the hyperbolic torsion. We prove such a derivative formula of the twisted
Alexander invariant for hyperbolic link exteriors like the Whitehead link
exterior. We provide the framework for the derivative formula to work, which
consists of assumptions on the topology of the manifold and on the
representations involved in the definition of the twisted Alexander invariant,
and prove derivative formula in that context. We also explore the symmetry
properties (with sign) of the twisted Alexander invariant and prove that it is
in fact a polynomial invariant, like the usual Alexander polynomial.
