How to determine the growth rate of a function on different intervals?

Question

Mathematics

In general, I found a description of the speed of functions on the Internet:
Original:

Do you want the value to grow slowly at first, but fast later? Use a polynomial or exponential function.
Do you want the value to grow fast at first, and slow down later? Use an nth-root or logarithmic function.

Translation:

SQRT(x) and logarithmic increase rapidly at first, but then slow down.
Power and exponential functions grow slowly at first, then accelerate.

I wanted to somehow prove these statements, but I do not know how. The main idea is to look at the second derivative, but I don’t know how to evaluate it. Take, for example, y = -x^2, y''= -2. This suggests that the speed of the derivative decreases all the time, but this function itself will be (-inf; 0) - increasing, (0; inf) - decreasing.
Things are not very good with the root either, where the second derivative is equal to (-1/4) * x^(-1.5). Which shows that this is an increasing function, and at infinity, it tends to zero. But how to prove that it increases sharply at first ....

Answer 1

The rate of change of a function is the first derivative. If it is greater than zero, then the function is growing; if it is equal to zero, then the function does not change in the vicinity of the given point; if it is less than zero, then the function is decreasing.
The larger the modulo derivative, the faster the function changes.

Answer 2

I would add to the original statement that this is true for monotonically increasing functions in the region from (0; inf), otherwise, as you yourself showed, examples that do not satisfy the statement are easily found.
In this formulation, it remains only to look at whether the second derivative is greater than zero or not.