Automatic differentiation (AD) is a technique for automatically augmenting
computer programs with statements for the computation of derivatives. This
article discusses the application of automatic differentiation to numerical
integration algorithms for ordinary differential equations (ODEs), in part
icular, the ramifications of the fact that AD is applied not only to the so
lution of such an algorithm, but to the solution procedure itself. This sub
tle issue can lead to surprising results when AD tools are applied to varia
ble-stepsize, variable-order ODE integrators. The computation of the final
time step plays a special role in determining the computed derivatives. We
investigate these issues using various integrators and suggest constructive
approaches for obtaining the desired derivatives.