Pardon the lack of rigor in my wording. I am looking for a way to quantify the rate of change of a function that compensates for the fact that this rate of change naturally slows down (in my example cases) when approaching the limit. I would like to know the name of such notion.
To illustrate, let's say I have a curve that raises towards a limit $L$ (say a function like $f(x)=L-Ae^{-\alpha x}$, $A>0, \alpha>0$). Imagine this function models for example the progress of some figure of merit of a technology towards a theoretical maximum. In many such cases, improving from $0.1L$ to $0.2L$ over a year is much easier (will take less research effort for example) than raising from $0.89L$ to $0.99L$ in the same amount of time. I am looking for an indicator that measures a rate of progress for that function, which "compensates" for the fact that progress becomes gradually harder as we get closer to the limit, so that it could reflect the "effort" needed, rather than the results.
To that effect, I define a "bounded progress" functional $BP[u]$, defined as $$BP[u]=\frac{1}{L-u}\frac{du}{dx}$$ Where the derivative is corrected by the first term which accounts for how close we are to the limit. Applied to the exponential example above, we would neatly get a constant value: $$BP[f]=\alpha$$ This comes as no surprise, because we note that this definition on BP, is equivalent to taking the derivative of $-log(L-u)$. The constant BP translates the general idea that, although the rate of change of this function is naturally decreasing as we approach the maximum, this curve can still reflect a "constant effort" in my case (just because progress becomes more "difficult").
I suppose this general idea must have been defined and used before, as I can see it being relevant in several fields of applied mathematics. My definition above is probably a naive one, and I am not looking for this one specifically, but rather for the name of this concept so that I can look into it and benefit from the experience of people who used it before.
Does anyone know a proper name for this notion?
Probably you just want the momentary progress wich is $f'(x)=A*\alpha*e^{-\alpha x}$ so if you put want the change when you change for 0,1 you have the change of f(x)$\Delta f=f'(x)*\Delta x$ in your case $\Delta x=0,1$