Let's say a company had these monthly sales:
- Jan: $10$
- Feb: $15$
- Mar: $22.5$
The growth rate of sales would be:
- Jan: -
- Feb: $50$%
- Mar: $50$%
What is the correct term for the "growth rate of the growth rate of sales"? e.g.:
- Jan: -
- Feb: -
- Mar: $0$%
There are no results found when googling "growth rate of growth rate", which leads me to believe I don't know the common term for how to refer to things like this.
I don't think there is an established term for the growth rate of a growth rate. However I wanted to point out that the growth rate of a function is not its derivative; $e^{kx}$ should have the constant growth rate $k$ to match with OP's example $10(1.5^n)$. The correct quantity should be the 'logarithmic derivative',
$$ \frac{df}{d\log x} = \frac{f'}{f}.$$
This matches with what is meant by 'growth rate of the economy', for instance.
(edit) At a cursory glance, I can't find anything about 'growth rates of a growth rate', or 'second order growth rate' or anything like that. But there is a name for the 'simplest' kind of function with non-zero 'second order growth rate', in the sense below. Suppose we had the relationships between functions $$ f(x) = e^{g(x)}, \quad g'(x) = C_0e^{h(x)}$$ Thus, the growth rate of $f$ is $g'$, and the 'second order growth rate' of $f$ is $h'$. You have discovered the case where $h' = 0$. The next simplest case is when $h'$ is a non-zero constant, so that $h(x) = Hx+C_1$ for some constants $H\neq 0, C_1\in\mathbb R$. By recentering in $x$, $C_1 = 0$. Then $g(x) = C_3e^{Hx}+C_2$, which then gives us: $$f(x) = e^{\displaystyle C_3 e^{Hx }+C_2} = C_4 \exp(C_3\exp Hx )$$
That is, these are the double exponential functions.