zlacker

> I was curious what it is called when the growth rate itself is going up exponentially

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.

>>wmanle+(OP)
There's two different ways to indicate how fast a function, F(x), is growing.

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.