The differentiator is made from a capacitor C, and resistor R, and assembled as shown.
A PULSE TRAIN is applied to the input.
When a pulse of voltage rises suddenly from zero to maximum, the current which is charging C suddenly rises to a maximum value as well.
As C charges, the charging current falls exponentially to zero.
Since this charging current is passing through R the voltage across R (which is the output voltage) does the same.***
Therefore we get the shape shown, with the voltage out rising suddenly to maximum and then falling exponentially to zero.

When the pulse falls to zero C discharges.
The discharge current is high at the start and then falls exponentially to zero as C discharges.
However, since the discharge current is in the opposite direction to the charge current the voltage across across R will be reversed and so the waveform is now shown below the zero line.

For each pulse the waveform out is repeated giving the display shown.

*** Ohms Law says that current is proportional to voltage. Conversely, voltage is proportional to current.