It was approximately exponential up until around 200ish AD, fell below exponential for a few hundred years, then was above exponential for around 600 years (the growth rate was going up approximately linearly), had a period where it varied and even was slightly negative, and then around 1500ish entered a period where the growth rate was increasing almost exponentially. That lasted to around 1960, and since then the growth rate rapidly.
Here's a graph of the growth rate from 4000 BC to 2023 [1] from the data here [2].
I was curious what it is called when the growth rate itself is going up exponentially, but utterly failed to craft a search in Google that worked for me. I then tried ChatGPT (the free version) and at first it was just wrong. I reiterated that I want to know what it is called when the growth rate is going up exponentially, not when the growth is exponential. It apologized and told me it is called "exponential growth of the growth rate" or "exponential acceleration".
I tried to verify that it is called "exponential acceleration" with Google, but failed.
The derivative of the exponential function is the exponential function.
d/dx eˣ = eˣ
So if it’s growing exponentially the rate of growth is exponential and the rate of that acceleration is also exponential.
One is to look at host much its value changes as x goes to x+d, and divide that to d to get an average rate of change from x to x+d. Take the limit as d -> 0 to get the instantaneous rate of change at x.
That gives a rate of change of Limit as d->0 of (F(x+d) - F(x)) / d, which is pretty much the textbook definition of d/dx F(x).
The other way is to look not at the actual value of the change but rather how much of a fraction of F(x) it was. That gives this measure: ((F(x+d)/F(x) - 1) / d. The instantaneous value would be the limit as d -> 0. That limit is of the form 0/0, but using L'Hôpital's rule we can turn it into (using the notation F'(...) for d/dx (F...)) the limit as d -> 0 of F'(x+d) / F(x) which is F'(x) / F(x).
When people talk of growth rate they usually mean this second measure. The first is usually called the rate of change. BTW, note that rate of change and growth rate are related. The growth rate is the rate of change of log(F(x)).
Exponential functions have an exponential rate of change but a constant growth rate. It is that constant growth that makes the concept of a half-life work for things that exponentially decay.