Hello, so I have a problem here - the only information I know about two functions f(x) and g(x) is the following:

f'(x)*g(x) = f(x)*g'(x) and g(x) cannot equal 0 on the interval (a,b)

What can we say about these two functions and how they are related?

Here is my thought process so far:

I'm thinking this can be split into two cases, where f'(x) = f(x) and g(x) = g'(x) OR where f'(x) = g'(x) and g(x) = f(x)

Then, from there we can say that in the first case, f(x)=g(x)=f'(x)=g'(x) which equals either 0 or e^x, since this is the only function where the derivative can equal itself

For the second case, we can say that it will be the same function

Is my reasoning correct? Am I missing something else about they how they may be related? Thanks