Differentiable simulators have recently shown great promise for training autonomous vehicle controllers. Being able to backpropagate through them, they can be placed into an end-to-end training loop where their known dynamics turn into useful priors for the policy to learn, removing the typical black box assumption of the environment. So far, these systems have only been used to train policies. However, this is not the end of the story in terms of what they can offer. Here, for the first time, we use them to train world models. Specifically, we present three new task setups that allow us to learn next state predictors, optimal planners, and optimal inverse states. Unlike analytic policy gradients (APG), which requires the gradient of the next simulator state with respect to the current actions, our proposed setups rely on the gradient of the next state with respect to the current state. We call this approach Analytic World Models (AWMs) and showcase its applications, including how to use it for planning in the Waymax simulator. Apart from pushing the limits of what is possible with such simulators, we offer an improved training recipe that increases performance on the large-scale Waymo Open Motion dataset by up to 12% compared to baselines at essentially no additional cost.