the differentiator
Swap the input resistor of an inverting amplifier for a capacitor and you get the integrator's mirror image: the output responds to how fast the input is changing, not to its level. Hold the input steady and the output is zero; change the input quickly and the output jumps. It reacts to slopes, ignoring constants.
The mechanism: a capacitor passes a current equal to C times the rate of change of the voltage across it, C times dVin over dt. That current flows into the virtual ground and through the feedback resistor R, so Vout equals minus RC times dVin over dt. A ramp rising at 2 V per second with RC of 0.1 second gives a steady minus 0.2 V; a sudden step gives a sharp spike.
Differentiators are used for edge or slope detection and for rate feedback in controllers, but they come with a real danger: their gain rises with frequency, so they amplify high-frequency noise and can easily become unstable and oscillate. Practical designs add a small resistor in series with the input capacitor and a small capacitor across the feedback resistor to roll off the high-frequency gain. For this reason the differentiator is used far less often than the integrator.
Feed a pulse train into a differentiator. Each rising edge produces a sharp positive-then-negative spike and each falling edge the opposite, marking exactly where the input changes fastest.
The differentiator picks out edges, the moments of fastest change.
A pure differentiator's gain rises with frequency, so it amplifies noise and can oscillate. Real designs add a series resistor and a parallel capacitor to limit high-frequency gain.