An Op-Amp Differentiator is a circuit that uses a operational amplifier, a capacitor and a resistor to calculate the derivative of the input signal. Its output signal is proportional to differentiation of input signal. The circuit for it is illustrated in the following photo.
If we do a transient analysis on this ciruict by equating the current across the capacitor to the current flowing through the resistance, we can get this first differential equation. Notice that the non-inverting terminal is grounded, the potential at the inverting terminal between the capacitor and resistor is also at 0 volts. After isolating the output voltage, we can see that it is directly proportional to the derivative of the input voltage.
We can also do an AC responce analysis, this time, we only use the impedance of the components. What we would find is that the circuit could also act as an high pass filter. This is because when the frequency is zero, the output voltage is also zero and when the frequency goes to infinity, the output goes to negative infinity, This means that the circuit has significant attenuation for low frequency voltage inputs and is therefore a high pass filter.
