Summary: We extend Darwiche's differential approach to inference in Bayesian Networks (BNs) to handle specific problems that arise in the context of Dynamic Bayesian Networks (DBNs). We first summarize Darwiche's approach for BNs, which involves the representation of a BN in terms of a multivariate polynomial. We then show how procedures for the computation of corresponding polynomials for DBNs can be derived. These procedures permit not only an exact roll-up of old time slices but also a constant-space evaluation of DBNs. The method is applicable to both forward and backward propagation, and it does not presuppose that each time slice of the DBN has the same structure. It is compatible with approximative methods for roll-up and evaluation of DBNs. Finally, we discuss further ways of improving efficiency, referring as an example to a mobile system in which the computation is distributed over a normal workstation and a resource-limited mobile device.