# Effect of differentiation on function growth rate

For sufficiently "nice" functions, the differentiation operator appears to make slow growing functions grow slower and fast growing functions grow faster, with $e^x$ as a fixed point in the middle.

My question is, how can we make this statement precise while still retaining as much generality as possible?

Clearly, there are all sorts of oscillatory and fractal-like functions for which the statement does not hold, which we must exclude in some manner. This feels similar to the Gronwall inequality, but I'm having trouble making a precise connection.