A generalization of the product rule for expressing arbitrary-order derivatives of products of functions, d^n/(d x^n)(u v) = (d^n u)/(d x^n) v + (n 1)(d^(n - 1) u)/(d x^(n - 1)) (d v)/(d x) + ... + (n r)(d^(n - r) u)/(d x^(n - r)) (d^r v)/(d x^r) + ... + u(d^n v)/(d x^n). where (n k) is a binomial coefficient. This can also be written explicitly as (D^~)^n f(t) g(t) = sum_(k = 0)^n(n k)(D^~)^k f(t)(D^~)^(n - k) g(t) (Roman 1980), where D^~ is the differential operator.

derivative | Faà di Bruno's formula | product rule