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I always use an op-amp configured as a differentiator. 741 works for me, but it might not be responsive enough for some applications. If I have a scope handy, I'll test it and send a sine wave into it and watch for a constant phase shift on the output. If you can get an eval copy of MicroSim's PSpice, you can simulate a lot of op amps pretty well in my opinion. It saves a ton of time.
Last time I used an integrator, I wanted a digital output and just tied an A-to-D converter to the output. 'Had some scaling problems, but it worked.
Rate of change is nothing more than the number of units that have changed divided by the time it took the change to occur. Yes, you can use an oscilloscope to measure that. For instance, the slew rate of an amplifier (how fast will the output change with a step change at the input) is a typical rate of change specification. Connect the scope to the output, a square wave generator to the input, look at the output on the scope (it will appear as a ramp, depending upon the sweep speed) and for a determined voltage change, say four divisions at 2 volts/div setting for 8 volts, measure the time from those same two points. Divide the voltage by the time to get the units in volts/second.
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