  A characteristic feature of differential-algebraic equations is that one
needs to find derivatives of some of their equations with respect to time, as
part of so called index reduction or regularisation, to prepare them for
numerical solution. This is often done with the help of a computer algebra
system. We show in two significant cases that it can be done efficiently by
pure algorithmic differentiation. The first is the Dummy Derivatives method,
here we give a mainly theoretical description, with tutorial examples. The
second is the solution of a mechanical system directly from its Lagrangian
formulation. Here we outline the theory and show several non-trivial examples
of using the "Lagrangian facility" of the Nedialkov-Pryce initial-value solver
DAETS, namely: a spring-mass-multipendulum system, a prescribed-trajectory
control problem, and long-time integration of a model of the outer planets of
the solar system, taken from the DETEST testing package for ODE solvers.
