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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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.
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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Take f(x) = x². The derivative is zero at x = 0, but the function is not constant. Anyway, just say "if" instead of "only if" and everything becomes 100% correct. No counterexample, no exception, just true.
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I never meant to refer to f'(x_0) when saying the derivative is zero. I meant f'(x)=0 for all x
End of conversation
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