The Chain Rule (page 141) states that the derivative of Use Carathéodory’s definition...

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.