Chain rule
Third proof

Constantin Carathéodory 's alternative definition of the differentiability of a function can be used to give an elegant proof of the chain rule.

Under this definition, a function f is differentiable at a point a if and only if there is a function q, continuous at a and such that f (x) − f (a) = q (x)(x − a). There is at most one such function, and if f is differentiable at a then f ′(a) = q (a).

Given the assumptions of the chain rule and the fact that differentiable functions and compositions of continuous functions are continuous, we have that there exist functions q, continuous at g (a), and r, continuous at a, and such that,

and

Therefore,

but the function given by h (x) = q (g (x)) r (x) is continuous at a, and we get, for this a

A similar approach works for continuously differentiable (vector-)functions of many variables. This method of factoring also allows a unified approach to stronger forms of differentiability, when the derivative is required to be Lipschitz continuous, Hölder continuous, etc. Differentiation itself can be viewed as the polynomial remainder theorem (the little Bézout theorem, or factor theorem), generalized to an appropriate class of functions.
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