Question

From: The Theory of Single Variable Calculus (Chapter 9)

Please Explain to me in detail! Thanks!

8. (a) Suppose g(x) = f(x+c). Prove (starting from the definition) that g'(x) = f'(x+c). Draw a picture to illustrate this. To do this problem you must write out the definitions of g'(x) and f'(x + c) correctly. The purpose of Problem 7 was to convince you that although this problem is easy, it is not an utter triviality, and there is something to prove: you cannot simply put prime marks into the equation g(x) = f (x +c). To emphasize this point: (b) Prove that if g(x) = f(cx), then g'(x) = c.f'(cx). Try to see pictorially why this should be true, also. (c) Suppose that f is differentiable and periodic, with period a (i.e., f(x +a) = f(x) for all x). Prove that f' is also periodic.

Answer 1

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