When humans perform contact-rich manipulation tasks, customized tools are often necessary and play an important role in simplifying the task. For instance, in our daily life, we use various utensils for handling food, such as knives, forks and spoons. Similarly, customized tools for robots may enable them to more easily perform a variety of tasks. Here, we present an end-to-end framework to automatically learn tool morphology for contact-rich manipulation tasks by leveraging differentiable physics simulators. Previous work approached this problem by introducing manually constructed priors that required detailed specification of object 3D model, grasp pose and task description to facilitate the search or optimization. In our approach, we instead only need to define the objective with respect to the task performance and enable learning a robust morphology by randomizing the task variations. The optimization is made tractable by casting this 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. We also validate that the shapes discovered by our method help real robots succeed in these scenarios.