The Chain Rule (page 141) states that the derivative of 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
b. Since f is differentiable at g(x), there is a function F that is continuous at 0 and such
c. For the function f(g(x)) we have
d. Therefore, the derivative of f(g(x)) is as was to be proved.
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