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.