  We define a derivation operator on the paths of a smooth, degenerate
diffusion process by smoothing the projection of the Sobolev derivative with
respect to the final data of the process, where the projection operator is
defined by the lifting of the orthogonal projection onto the range of the
diffusion coefficient to the Cameron-Martin space. As application we prove
Clark formula, Poincar\'e and logarithmic Sobolev inequalities.
