  We apply the Dirichlet forms version of Malliavin calculus to stochastic
differential equations with jumps. As in the continuous case this weakens
significantly the assumptions on the coefficients of the SDE. In spite of the
use of the Dirichlet forms theory, this approach brings also an important
simplification which was not available nor visible previously : an explicit
formula giving the carr\'e du champ matrix, i.e. the Malliavin matrix.
Following this formula a new procedure appears, called the lent particle method
which shortens the computations both theoretically and in concrete examples.
