The only solution that I can think of is if $f(x) = c \times g(x)$, where $c \in \mathbb{R}$ but I think this is trivial. I've considered letting $f'(x) = g''(x)$ to make the LHS easy to integrate. While this leads to a difficult ODE, I think this is another solution. Any other solutions?
What functions $f(x)$ and $g(x)$ satsifies: $\frac{f'(x)}{g'(x)} = \left( \frac{f(x)}{g(x)}\right)'$
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1Check this: https://math.stackexchange.com/q/10927/42969. – Martin R Oct 01 '21 at 14:03
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This seems to be a duplicate, thank you! – Renaissance Oct 01 '21 at 14:05