For example, is this derivative useless: $$\lim_{h\rightarrow 0}\log_{\frac{x+h}{x}}\frac{f(x+h)}{f(x)}$$ It evaluates to $f^{'}(x)\cdot \frac{x}{f(x)}$, where $f^{'}(x)$ is the normal derivative of the function. I could think of another one: $$\lim_{h\rightarrow 0}\left(\frac{f(x+h)}{f(x)}\right)^{\frac{1}{h}}$$ This one evaluates to $e^{\frac{f^{'}(x)}{f(x)}}$ I had used these in approximations in some of my previous posts. But that's not a good use to talk about when there are so many other methods for that. Can you think of any other use of these?
Why aren't any derivatives defined to measure logarithmic change in a function?
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1https://en.wikipedia.org/wiki/Multiplicative_calculus are what you are thinking of, I think. In particular your second idea has a name, called the geometric derivative. – 2017-02-11