We propose the formulation of forward and differentiable rigid-body dynamics using Lie-algebra rotation derivatives. In particular, we show how this approach can easily be applied to incremental-potential formulations of forward dymamics, and we introduce a novel definition of adjoints for differentiable dynamics. In contrast to other parameterizations of rotations (notably the popular rotation-vector parameterization), our approach leads to painlessly simple and compact derivatives, better conditioning, and higher runtime efficiency. We demonstrate our approach on fundamental rigid-body problems, but also on Cosserat rods as an example of multi-rigid-body dynamics.