Given two functions f and g, the product rule tells you how to take the first derivative of their product, and the chain rule tells you how to take the first derivative of their composition. What if you want to take higher order derivatives? You could repeatedly apply basic calculus rules, but there are formulas for taking the higher order derivatives all at once.
The generalization of the product rule is known as Leibniz rule. It’s fairly simple and looks a lot like the binomial theorem.
The generalization of the chain rule is known as Faà di Bruno’s theorem. It’s more complicated, and it uses exponential Bell polynomials, something I’ve blogged about a few time lately.
More Bell polynomial posts: