Differentiator is an op amp based circuit, whose output signal is proportional to differentiation of input signal.
A differentiator is an op amp based circuit. This is basically an inverting amplifier with a capacitor of suitable value at its input terminal. The basic circuit diagram of a differentiator is shown in the figure below, differentiator circuit

The op amp used here in an ideal op amp. We know that the voltage at both inverting and non inverting terminals of an ideal op amp is same. As the electric potential at non inverting terminal is zero. Since, it is grounded, the electric potential of inverting terminal is zero, too. It is also known to us that the current entering through inverting and non inverting terminal of an ideal op amp are zero.
Considering, these conditions of ideal op amp, if we apply Kirchhoff Current Law at node 1 of the differentiator circuit, we get, The above equation shows the output voltage is the derivative of the input voltage.

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