  We consider a linear perturbation in the velocity field of the transport
equation. We investigate solutions in the space of bounded Radon measures and
show that they are differentiable with respect to the perturbation parameter in
a proper Banach space, which is predual to the H\"older space
$\mathcal{C}^{1+\alpha}(\mathbb{R}^d)$. This result on differentiability is
necessary for application in optimal control theory, which we also discuss.
