When humans perform contact-rich manipulation tasks, customized tools are often necessary to simplify the task. For instance, we use various utensils for handling food, such as knives, forks and spoons. Similarly, robots may benefit from specialized tools that enable them to more easily complete a variety of tasks. We present an end-to-end framework to automatically learn tool morphology for contact-rich manipulation tasks by leveraging differentiable physics simulators. Previous work relied on manually constructed priors requiring detailed specification of a 3D object model, grasp pose and task description to facilitate the search or optimization process. Our approach only requires defining the objective with respect to task performance and enables learning a robust morphology through randomizing variations of the task. We make this optimization tractable by casting it as a continual learning problem. We demonstrate the effectiveness of our method for designing new tools in several scenarios, such as winding ropes, flipping a box and pushing peas onto a scoop in simulation. Additionally, experiments with real robots show that the tool shapes discovered by our method help them succeed in these scenarios.