That expression is sometimes refered to the logarithmic derivative, because:
Since the logarithm grows very slowly, that expression - the rate of change of ln(|f(x)|) - should be "small" expect where a function is really "spiking" - growing or shrinking very quickly.
Consider, it's 1 for , and is for . So, up to a constant multiple, it's like even for a "hugely" increasing function like
For complex variables, that expression takes on a new and different significance.