Chain Rule This exercise proves the Chain Rule without the special assumption made in
Chapter 3, Problem 105(choose chapter or problem)
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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