We compute flow sensitivities by modifying a CFD code which is spatially-discretized with spectral differences. Our discrete approach relies on algorithmic differentiation. We obtain two transformed codes, one for each differentiation mode : tangent and adjoint. Both codes compute sensitivities in an unsteady test case of two-dimensional incompressible flow inside a periodic cube with an initial double-shear profile. The sensitivities from both codes agree to within machine accuracy, and compare well with those approximated by finite difference computations. We discuss execution times and describe our strategy to automatically differentiate in adjoint mode a parallel code containing MPI instructions.