Explanation Given: For any number b, define a new function F ( u ) = f ( u ) − f ( b ) u − b ∀ u ≠ b Show the following a) 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 ∀ u f ( u ) − f ( g ( a ) ) x − a = F ( u ) u − g ( a ) x − a → ( 2 ) c) Substitute u = g ( x ) in f ( u ) − f ( g ( a ) ) x − a = F ( u ) u − g ( a ) x − a to get f ( g ( x ) ) − f ( g ( a ) ) x − a = F ( g ( x ) ) g ( x ) − g ( a ) x − a Derive the chain rule by computing the limit of both sides as x → a Proof: For any differential function f and any number b, define F ( u ) = f ( u ) − f ( b ) u − b ∀ u ≠ b a. Define F ( b ) = f ' ( b ) , then lim u → b F ( u ) = lim u → b f ( u ) − f ( b ) u − b = f ' ( b ) = F ( b ) ⇒ lim u → b F ( u ) = F ( b ) Therefore, F is continuous at u = b b. Let g be a differentiable function and take b = g ( a ) . Let x be a number distinct from a. If we substitute u = g ( a ) into equation (2), both sides evaluate to 0, so equality is satisfied

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Question

Chapter 3.7, Problem 105E

To determine

To prove:

The chain rule without the special assumptions

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Chapter 3.7, Problem 104E

Chapter 3.8, Problem 1PQ

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Chapter 3 Solutions

Calculus - Standalone book

Ch. 3.1 - Prob. 1PQ Ch. 3.1 - Prob. 2PQ Ch. 3.1 - Prob. 3PQ Ch. 3.1 - Prob. 4PQ Ch. 3.1 - Prob. 5PQ Ch. 3.1 - Prob. 1E Ch. 3.1 - Prob. 2E Ch. 3.1 - Prob. 3E Ch. 3.1 - Prob. 4E Ch. 3.1 - Prob. 5E

Ch. 3.1 - Prob. 6E Ch. 3.1 - Prob. 7E Ch. 3.1 - Prob. 8E Ch. 3.1 - Prob. 9E Ch. 3.1 - Prob. 10E Ch. 3.1 - Prob. 11E Ch. 3.1 - Prob. 12E Ch. 3.1 - Prob. 13E Ch. 3.1 - Prob. 14E Ch. 3.1 - Prob. 15E Ch. 3.1 - Prob. 16E Ch. 3.1 - Prob. 17E Ch. 3.1 - Prob. 18E Ch. 3.1 - Prob. 19E Ch. 3.1 - Prob. 20E Ch. 3.1 - Prob. 21E Ch. 3.1 - Prob. 22E Ch. 3.1 - Prob. 23E Ch. 3.1 - Prob. 24E Ch. 3.1 - Prob. 25E Ch. 3.1 - Prob. 26E Ch. 3.1 - Prob. 27E Ch. 3.1 - Prob. 28E Ch. 3.1 - Prob. 29E Ch. 3.1 - Prob. 30E Ch. 3.1 - Prob. 31E Ch. 3.1 - Prob. 32E Ch. 3.1 - Prob. 33E Ch. 3.1 - Prob. 34E Ch. 3.1 - Prob. 35E Ch. 3.1 - Prob. 36E Ch. 3.1 - Prob. 37E Ch. 3.1 - Prob. 38E Ch. 3.1 - Prob. 39E Ch. 3.1 - Prob. 40E Ch. 3.1 - Prob. 41E Ch. 3.1 - Prob. 42E Ch. 3.1 - Prob. 43E Ch. 3.1 - Prob. 44E Ch. 3.1 - Prob. 45E Ch. 3.1 - Prob. 46E Ch. 3.1 - Prob. 47E Ch. 3.1 - Prob. 48E Ch. 3.1 - Prob. 49E Ch. 3.1 - Prob. 50E Ch. 3.1 - Prob. 51E Ch. 3.1 - Prob. 52E Ch. 3.1 - Prob. 53E Ch. 3.1 - Prob. 54E Ch. 3.1 - Prob. 55E Ch. 3.1 - 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To prove: The chain rule without the special assumptions

Solution Summary: The author explains the chain rule without the special assumptions. For any differential function f and any number b, define a new function F(u)=f' (b