  Gradient estimates are derived, for the first time, for the semigroup
associated to a class of stochastic differential equations driven by
multiplicative L\'evy noise. In particular, the estimates are sharp for
$\alpha$-stable type noises. To derive these estimates, a new derivative
formula of Bismut-Elworthy-Li's type is established for the semigroup by using
the Malliavin calculus and a finite-jump approximation argument.
