I've been messing around with exponentials lately, and an interesting question I asked myself was, is it possible to calculate the exponential growth at a point?
So, I'm sure you're familiar with derivatives and tangents. With a derivative if you have $f(x)$, and $f'(x)$ is constant from $a$ to $a+1$, then $f(a+1) = f(a)+f'(a)$
Now, let's say we have some function $f(x)=2^x$ where the rate of growth is constant(100% growth every unit), now let's say $g(x)$ will be our equation to describe exponential growth at each point. Since it's constant throughout, $g(x)=2$ at all times. You can then say that, if $g(x)$ is constant from $a$ to $a+1$, then $f(a+1) = f(a)*g(a)$
I've been able to calculate that $$g(x)=exp\left(\frac{f'(x)}{f(x)}\right)$$ Now, this is very interesting, and even somewhat useful to me especially for regression, since exponential growth rates do in fact change a lot in the real world. And you can either solve $g(x)=exp\left(\frac{y'}{y}\right)$ as a differential equation on a case by case basis to get the exponential for a specific growth function, or use the more general integral, particularly if you want to compute it numerically: $$f(x)=c_1exp\left(\int_1^xln(g(t))dt\right)$$
Explanation aside, my question is does $g(x)$ have a more formal name, and where are some places where I can learn more about, or see such math being used?
Edit: After a brief discussion in the comments, I have realized I need to make more clear what exactly g(x) is doing, so I will write, in order, that $$g(x)=\lim_{h\to 0} \sqrt[h]{\frac{f(x+h)}{f(x)}} = exp\left(\frac{f'(x)}{f(x)}\right)$$ And that in a similar to which $f(x+h)\approx f(x)+f'(x)*h$, $$f(x+h)\approx f(x)g(x)^h$$
After discussing what the operation I was doing really meant in the comments, a new name hit me, multiplicative derivative, and after a quick google search, it turns out the exact thing I have described does in fact already exist and has been written about! It has been coined the term "Multiplicative calculus" by others, and many papers seem to have been written with it. I will post just one, below. The symbol commonly used for it is $$f^*(x)=\lim_{h\to 0}\left(\frac{f(x+h)}{f(x)}\right)^{\frac{1}{h}}=e^{\frac{f'(x)}{f(x)}}$$
similar to how I had already described. Regardless of ending up just answering my own question, hopefully this can serve as a gateway for anyone who was also curious of similar things to find the term they may have been looking for, and thanks to @memerson in particular for our discussion in the comments, which led to me discovering the name.
https://core.ac.uk/download/pdf/81954511.pdf