Small resistor value for a differentiator
Your differentiator circuit can also be thought of as a High Pass Filter. This might be a little easier to understand.

There is a Corner Frequency for the High Pass Filter which is;
F  =  1 / (2 Pi R C)

At the Corner Freq  R = Xc  where Xc = 1 / (2 Pi F C)

At frequencies below the Corner Freq. the response is increasing at 6 db/octave.

As long as there is this increasing response then the circuit is acting like a Differentiator.

Do you use Spice? Such as LTSpice?

It is free and pretty easy to use.

You can put your circuit in there and work with it.

For your original question, just pick a resistor, say 1000 ohms and then compute a capacitor using the above 2 formulas, such that the response is always changing at +6db/octave in the region where you want to "differentiate a signal".

If you haven't used Spice (such as LTSpice), I think you might find it a lot of fun and in the future, indispensable.

Here is a link for Windows version of LTSpice;
LTSpice for Windows CLICK HERE

All the Best,
xstack