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What is the largest class of everywhere differntiable real functions of one variable such that the product of the derivatives is the derivative of the product? Certainly the constant functions satisfy my conditions, but is it the largest class of functions?

user107952
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1 Answers1

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Here is an example. Let f(x) = $e^{ax}$ and g(x) = $e^{bx}$. Then f'(x)g'(x) = $(ab)e^{(a+b)x}$ and d/dx(fg) = $(a+b)e^{(a+b)x}$ .

So you have equality when ab = a + b, which comes down to b = a/(a-1).

I imagine you can do a similar computation for the exponential functions of any base.

Whether there are other classes of functions that might work I do not know.

Betty Mock
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  • So, there is no largest class of functions satisfying my conditions. – user107952 Dec 02 '13 at 01:37
  • @user107952 Well, I'm not in a position to say that. – Betty Mock Dec 02 '13 at 18:58
  • Actually, there really isn't. The class {e^2x} is one such class, and so are the class of all constant functions. But if you unite those classes, they no longer satisfy my condition. So, there isn't a largest class. Perhaps a better question would be, is there a largest class of functions satisfying my conditions, and also contains all constant functions? If so, what is it? – user107952 Dec 02 '13 at 22:35
  • I agree that combining the exponentials and constants would not work. However, you do have yet another set of functions -- those whose products are constant. – Betty Mock Dec 04 '13 at 06:00