Has There Been Any Study of the Quotient-root Derivative Definition $\lim_{\Delta x \to 0} \sqrt[\Delta x]{\frac{f(x + \Delta x)}{f(x)}}$?

Question

Has there been a study of turning the difference-quotient seen in the common derivative $$ \frac d {dx} f(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x} $$

into a quotient-root $$ Д_x f(x) = \lim_{\Delta x \to 0} \sqrt[\Delta x]{\frac{f(x + \Delta x)}{f(x)}}? $$

By approximating the function in Desmos, it seems that: \begin{align} Д_x C & = 1 \\ Д_x k f(x) & = Д_x f(x) \\ Д_x x^n & = e^\frac{n}{x} \\ Д_x a^x & = a \\ Д_x x^x & = ex \end{align}

to list a few general functions.

Would such a definition allow for better definitions of hyperoperations like tetrations? Also, how would an integral calculus look with this as the basis? Are there any other types of derivatives out there like this? Can this be expressed with Leibniz notation?

Answer 1

$$y=\left(\frac{f(x+\epsilon )}{f(x)}\right)^{\frac{1}{\epsilon }}$$

Take logarithms, expand $f(x+\epsilon)$ as a series around $\epsilon=0$ to obtain $$\log(y)=\frac{f'(x)}{f(x)}+\frac{ f(x) f''(x)-f'(x)^2}{2 f(x)^2}\epsilon+O\left(\epsilon ^2\right)$$ which makes the limit to be $$y=\exp\Bigg[\frac{f'(x)}{f(x)} \Bigg]$$