While models in audio and speech processing are becoming deeper and more end-to-end, they as a consequence need expensive training on large data, and are often brittle. We build on a classical model of human hearing and make it differentiable, so that we can combine traditional explainable biomimetic signal processing approaches with deep-learning frameworks. This allows us to arrive at an expressive and explainable model that is easily trained on modest amounts of data. We apply this model to audio processing tasks, including classification and enhancement. Results show that our differentiable model surpasses black-box approaches in terms of computational efficiency and robustness, even with little training data. We also discuss other potential applications.