The Freshman's Product Rule (f⋅g)' = f'⋅g' is true only if f/f' + g/g' = 1pic.twitter.com/6eHFtV7BwI
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It is, at x = 1. And I am evaluating at the same x. You don't divide by zero because the pole is cancelled out. But anyway, my point is that the logic is reversed. You're using "p only if q" (p => q), when it should be "p if q" (p <= q).
I'm not sure about that, the proof goes both ways. Suppose f and g are non-constant functions. It follows thatpic.twitter.com/MtPxH6xVh0
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I just gave you a counterexample! If you desperately want to keep the "only if", then you have to restrict the statement to f' ≠ 0, g' ≠ 0. Also, f' = 0 doesn't imply that f is constant. Even over an open domain. Take the Cantor function over the complement of the Cantor set.
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I did say the derivatives were nonzero when I said f,g were not constant... if you want to go digging in Rudin for pathological functions, be my guest.. but I'm not going to do that for every thing I tweet.. I will always refer to non-pathological functions
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multiply by f'g' in the first line, and divide by f'g' in the second
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