What condition must be verified by the frequency so that an RC circuit acts as a differentiator?
The condition is defined  by time domain exponential behavior and frequency domain.

You requirements  must include some specification to be below some cutoff frequency   or exponential constant T=RC with some tolerance such T defines the voltage decay to 64% of target.

e.g. differentiator for f &lt; fo/10 or 20 dB down  or T=1/fo=1/2piRC

The current phase leads voltage by 90 degrees.

e.g. a High Pass Filter performs differentiation (to some tolerance) below fo/10 typ.

A ideal differentiator has no upper limit in frequency gain +6dB/octave.

Other equivalent conditions  to + 6dB/octave are +20dB/octave for the frequency range of interest.

passive methods

a) 
It can  block DC current with a series C with a shunt R
b) or shunt DC current to 0 volts with a series R and a shunt L.