Does the formula $\lim_{\Delta x \rightarrow 0} \sqrt[\Delta x]{\frac{f(x+\Delta x)}{f(x)}} = e^{\frac{f'(x)}{f(x)}}$ have a name? I have derived the following formula for a project, and I have not been able to find it anywhere online. Is there a name for this formula? Has anyone seen it in the literature?
$$\lim_{\Delta x \rightarrow 0} \sqrt[\Delta x]{\frac{f(x+\Delta x)}{f(x)}} = e^{\frac{f'(x)}{f(x)}}$$
If I were to come up with a name for it, I would call it the “exponential derivative” of the function $f(x)$, because it is similar to the definition of a derivative, but instead of subtracting two values of the function we are dividing, and instead of dividing by $\Delta x$ we are taking a root. In other words, this formula considers the exponential change in the function $f(x)$, rather than the linear change in the case of a regular derivative. However, if this formula already has a name, it would be helpful to know how to refer to it. Thanks!
 A: As Essaidi and Lorago pointed out, $f^*(x) := \exp\left(\frac{\mathrm{d}}{\mathrm{d}x}\ln f(x)\right) = e^{f'(x)/f(x)}$ is called the multiplicative/geometric derivative.
Your intuition was almost correct : the usual derivative $f'$ measures the linear (additive) change  of $f$ with respect to $x$, whereas $f^*$ captures its variation in a geometric (multiplicative) way.
Note that its multiplicative nature implies the nice product rule $(fg)^* = f^*g^*$. In fact, in a more general context, you can define new derivatives $D_\varphi = \varphi \circ \frac{\mathrm{d}}{\mathrm{d}x} \circ \varphi^{-1}$ in order to design a desired property; in the present case, the linearity of $\frac{\mathrm{d}}{\mathrm{d}x}$ is mapped to this new product rule, because $\varphi = \exp$ is an isomorphism between the groups $(\mathbb{R},+)$ and $(\mathbb{R}^\times,\cdot)$.
Finally, it is to be noted that the inverse operator of the multiplicative derivative is the product integral (see e.g. https://en.wikipedia.org/wiki/Product_integral) $$\prod_a^b f(x)^{\mathrm{d}x} = \exp\left(\int_a^b \ln f(x) \mathrm{d}x\right) = \frac{F_*(b)}{F_*(a)},$$ where $F_*$ is the multiplicative antiderivative of $f$.
