  The notion of derivatives for smooth representations of $GL(n,\mathbb{Q}_p)$
was defined in [BZ77]. In the archimedean case, an analog of the highest
derivative was defined for irreducible unitary representations in [Sah89] and
called the "adduced" representation. In [AGS] derivatives of all orders were
defined for smooth admissible Frechet representations (of moderate growth).
  A key ingredient of this definition is the functor of twisted coinvariants
with respect to the nilradical of the mirabolic subgroup. In this paper we
prove exactness of this functor and compute it on a certain class of
representations. This implies exactness of the highest derivative functor, and
allows to compute highest derivatives of all monomial representations.
  In [AGS] these results are applied to finish the computation of adduced
representations for all irreducible unitary representations and to prove
uniqueness of degenerate Whittaker models for unitary representations.
