Carathéodory’s Definition of the Derivative and Proof of the Chain Rule
The Chain Rule (page 155) states that the derivative of f ( g(x)) is f ‘( g(x)) · g’(x). Use Carathéodory’s definition of the derivative to prove the Chain Rule by giving reasons for the following steps.
a. Since g is differentiable at x, there is a function G that is continuous at 0 and such that g(x + h) - g(x) = G(h) · h, and G(0) = g’(x).
b. Since f is differentiable at g(x), there is a function F that is continuous at 0 and such that f(g(x) + h) - f(g(x)) = F(h) · h, and F(0) = f’(g(x)).
c. For the function f(g(x)) we have
d. Therefore, the derivative of f(g(x)) is F(g(x + 0) - g(x)) · G(0) = F(0) · G(0) = f’(g(x)) · g’(x) as was to be proved.
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