Chain Rule This exercise proves the Chain Rule without the special assumption made in

Chapter 3, Problem 105

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Chain Rule This exercise proves the Chain Rule without the special assumption made in the text. For any number b, define a new function F (u) = f (u) f (b) u b for all u = b (a) Show that if we define F (b) = f (b), then F is continuous at u = b. (b) Take b = g(a). Show that if x = a, then for all u, f (u) f (g(a)) x a = F (u)u g(a) x a Note that both sides are zero if u = g(a). (c) Substitute u = g(x) in Eq. (2) to obtain f (g(x)) f (g(a)) x a = F (g(x)) g(x) g(a) x a

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