A way to organize functions by their speed of growth ?

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A way to organize functions by their "speed of growth"?

How does one say formally in math that a certain function grows "faster" than another?
Doens't really work for trig functions, i know.
you knotice that the exponential function is the function dividing d/dx slower than itself and d/dx faster than itself functions

In order from slower to faster:

Derivative is slower than itself:
Constants

rational function in which quotient is non-constant

Logarithms

Roots

Non-constant polynomials:

b^x, b>1: derivative is proportional to itself

Derivative is faster than itself:

Self-power:x^x

Gamma(x)

Tetrational function [tex]{}^xb=b[4]x[/tex]; b[4]1=b, b[4]2=b^b, b[4]3=b^(b^b), etc. (note the grouping)
 
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