Reinforcement learning treats the plant as a black box: it perturbs actions, observes rewards, and estimates which way to nudge the policy — needing thousands to millions of interactions because it has no gradient, only samples. Differentiable control throws that away. When you have a differentiable model of the plant — analytic, or one of the learned models from this course — the entire closed-loop rollout is a differentiable function of the controller's parameters, so one backward pass gives the exact gradient of the trajectory cost, and gradient descent tunes the controller directly. The community reports differentiable model-predictive control reaching good policies with roughly an order of magnitude fewer samples than RL. Here the controller is a PID, tuned by rolling the loop out on the autodiff tape with the gains as variables; the cost weights setpoint error over the trajectory plus a little control effort. Two things to notice: it settles on the setpoint, and it gets there by finding that Ki must be positive — the gradient discovered the need for an integrator, the same fact a controls engineer knows, learned from data. This is 'PID tuned by backpropagation,' and it generalizes: swap the plant for a learned model and you can train a neural controller the same way, which is the capstone.