  For a semigroup $P_t$ generated by an elliptic operator on a smooth manifold
$M$, we use straightforward martingale arguments to derive probabilistic
formulae for $P_t(V(f))$, not involving derivatives of $f$, where $V$ is a
vector field on $M$. For non-symmetric generators, such formulae correspond to
the derivative of the heat kernel in the forward variable. As an application,
these formulae can be used to derive various shift-Harnack inequalities.
