tbhausen what’s considered “large” for a first derivative?
OK, if the slope is x arcseconds/second of time, then a guide star would move x arcseconds while you are taking a 1-second guide exposure! Check out why it cannot be less by looking out for the "sawtooth" graphs that I have posted here. Once you grok that single graph, things should become clear (just like the Claritin TV commercials :-)
So, if you are using a 2 second (0.5 FPS) guide exposure, and x is 0.25 arcsecond/second, then the guide star(s) would have moved (at the times when the slope is maximum at rate x) by 0.5 arcseconds.
The centroid calculation would need to be able to measure this 0.5 arcsecond trailing -- and worse, is that by the time it measures the centroid, the star is already somewhere else 0.25 arcseconds away -- and that is just from one axis of your mount (think Wayne Gretzky :-) This makes guiding to better than 0.5 arcsecond RMS virtually unmöglich.
So, the trick is to use shorter exposure time, so that the guide star has no chance to move much, given the same x arcsecond/second "slope."
So, in answer to your question of what a large first derivative is, you want x to be small enough so that the guide star trailing is small for the exposure time that you use. If x is large, you will need to compensate with a shorter exposure time. But at some point, your guide scope/software combination can no longer handle even shorter exposure times (run out of photons, or processor speed to calculate the centroid, or the command protocol latency to the mount is too large, etc). That is when x is too large.
x of 0.25 arcsecond/second with 0.5 second guide exposure time should let you get to better than 0.5" type total RMS error, if you are careful, and if there are no machining imperfections that is not directly related to the gears. So, assuming there are no mechanical glitches, you should be able to do something in the region of 0.4" total RMS error on a windless day (or in between the wind gusts). I have my mount (albeit, not a ZWO mount, but it is a strain wave gear mount) dialed down to the 0.35" to 0.4" region, and can actually see my guide graph move by more than 0.5 arcsecond each time I hear the leaves (I have trees all around me -- this is tree-hugging country :-) rustle outside with just a 5 mph gust (I don't have a dome).
And yes, you are right. A 287 second period will automatically increase the slope from a 430.8 second period by a 50%, if all else were equal. However, if the 287-second mount has 50% smaller peak-to-peak error, then the slopes will again be equal.
By the way, I use 287 and 430.82 seconds because the 288 and 432 numbers are for sidereal time. The hour angle of your mount moves one revolution every 23 hours 56 minutes, not every 24 hours -- a target in the sky rises 4 minutes earlier each passing day.
The second technique (I only discovered this by serendipity after I had stared at the "sawtooth" graph myself one day, and it suddenly hit me) is to slow down the guide rate. Once I had a clear night after the realization, I tried it, and it worked as predicted (for me). Give it a shot. It really helps as long as the "x" is small enough to allow it, and if your guide software allows it (ASIAIR does not allow it for ZWO mounts; some infinite wisdom from probably the same person who defaulted ASIAIR max pulse durations to 2000 ms).
Finally, measuring that piece of paper from ZWO only gives you a lower bound of "x". The graph does not show the places where the slope is largest (ZWO still think that slope is not critical, but peak-to-peak error is). So, there could be hour angles where "x" is actually larger.
The way you can tell is to apply the max pulse duration from the "x" that you measure. If some time during the night (i.e., at some hour angle of the motor) you see a clipped comb of RA (or declination) pulses, then the max duration setting that you got from "x" is not large enough to handle the actual slope at that moment. It is really easy to spot since you have this long sequence of pulses (a second to a few seconds) on your guide graph, all with the same amplitude (that is the amplitude of the max pulse). Make sure it is not some external influence (wind, cat or deer walking by, etc) that causes the comb.
BTW, if the sawtooth amplitude is 0.5 arcseconds on your guide camera, your main camera will also see precisely (unless there is some flexure) the same sawtooth.