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Recent differentiable rendering techniques have become key tools to tackle many inverse problems in graphics and vision. Existing models, however, assume steady-state light transport, i.e., infinite speed of light. While this is a safe assumption for many applications, recent advances in ultrafast imaging leverage the wealth of information that can be extracted from the exact time of flight of light. In this context, physically-based transient rendering allows to efficiently simulate and analyze light transport considering that the speed of light is indeed finite. In this paper, we introduce a novel differentiable transient rendering framework, to help bring the potential of differentiable approaches into the transient regime. To differentiate the transient path integral we need to take into account that scattering events at path vertices are no longer independent; instead, tracking the time of flight of light requires treating such scattering events at path vertices jointly as a multidimensional, evolving manifold. We thus turn to the generalized transport theorem, and introduce a novel correlated importance term, which links the time-integrated contribution of a path to its light throughput, and allows us to handle discontinuities in the light and sensor functions. Last, we present results in several challenging scenarios where the time of flight of light plays an important role such as optimizing indices of refraction, non-line-of-sight tracking with nonplanar relay walls, and non-line-of-sight tracking around two corners.