Approximation of a Non Ideal Differentiator
Here's a less mathematically rigorous but hopefully easy to grasp way of thinking about this.

What you have from the left end of C1 to the output is a pure differentiator.  The only thing that makes the circuit not a differentiator is R1.  R1 working against C1 is a low pass filter, which undoes the effect of the differentiator.

This circuit is therefore a differentiator when R1 is "small" compared to C1.  Put another way, the differentiator part of this circuit dominates below the frequency where the R1-C1 low pass filter becomes significant.

The frequency rolloff of a R-C filter is 1/2&Pi;RC.  In your case, at frequencies well below that, the low pass filter is effectively not there in that it simply passes its input signal to the output.  At frequencies well above the rolloff, the attenuation is proportional to the frequency divided by the rolloff frequency.

The answer as to what frequency range this circuit acts like a differentiator depends on what error you can tolerate.  The R-C filter action never completely goes away with lower frequency.  It only approaches that.  Only you can say how little effect is little enough.