Ribonucleic acid (RNA) is an essential molecule in a wide range of biological functions. In 1990, McCaskill introduced a dynamic programming algorithm for computing the partition function of an RNA sequence. This forward model is widely used for understanding the thermodynamic properties of a given RNA. In this work, we introduce a generalization of McCaskill's algorithm that is well-defined over continuous inputs and is differentiable. This allows us to tackle the inverse folding problem---designing a sequence with desired equilibrium thermodynamic properties---directly using gradient optimization. This has applications to creating RNA-based drugs such as mRNA vaccines. Furthermore, it allows McCaskill's foundational algorithm to be incorporated into machine learning pipelines directly since we have made it end-to-end differentiable. This work highlights how principles from differentiable programming can be translated to existing physical models to develop powerful tools for machine learning. We provide a concrete example by implementing an effective and interpretable RNA design algorithm.