I am reading Mandelbrot, and stubling upon his use of the limit ("almost a Hölder exponent") $$\lim_{\varepsilon\to0}\frac{\log(f(x+\varepsilon) - f(x))}{\log(\varepsilon)}.$$

To simplify, lets assume that $f$ is non-decreasing, and of course the limit of $\epsilon$ is from the right. Calling this (me, not Mandelbrot!) the "log-derivative", we can calculate log-derivative $x^\alpha$ at $x=0$ is $\alpha$ ($\alpha>0$). For a differentiable function, such that the first derivative is positive, we can calculate (by Taylor) the log-derivative is $1$, and if the function is almost a constant, such that many derivatives are zero, and k being the least integer such that the $k$th derivative is positive, the log-derivative is $k$. And so on. The idea of course is that this will be defined, at least for some non-differentiable function.

My question: Does this procedure have an official name? any references? It is difficult to google because log-derivative means something else ...