The Chain Rule for Higher Derivatives

Perhaps the most important theorem of elementary differential calculus is the Chain Rule. It states, roughly, that the composite of two differentiable functions is again differentiable, and it gives a formula for the derivative of this composite. A Chain Rule of Order n should state, roughly, that the composite of two functions that are n times differentiable is again n times differentiable, and it should give a formula for the n’th derivative of this composite. Many textbooks contain a Chain Rule of order 2 and perhaps 3, but I do not know of a single one that contains a Chain Rule of arbitrary order n with an explicit and useful formula. The main purpose of this paper is to derive such a formula.