you might want to ask your self, what does the bode plot of a differentiator look like.... you are really frustrating. look at any standard differentiator circuit or even a high pass filter circuit, there is a cut off frequency at some point.
https://upload.wikimedia.org/wikipedia/commons/6/63/Bode_High-Pass.PNG
it acts as an differentiator in lower frequencies... the cut off frequency is when it starts turning into a flat band.
Why would I ask mysefl that, then look at a bode plot of a high-pass filter? Here is a real bode of a diffeentiator.
Read Differentiator Bode