I have two, A Davis .22 Rimfire that accompnies me on many field trips just for snakes and feral dogs, and a Cobra .38 Special. I have had a friendload some special low velocity handloads to play with. These are about the equivilant of a ..38 Specoal CB CAP! Standard .38 Specials are a handful due to the small grips, you don't get much purchase on the gripframe!
The real secret on shooting a modern Derringer is to pull down on the trigger as the instruction book says to! The second thing about the triggers is that they need to be broken-in by shooting the gun. burrs and rough contact surfaces smooth out with use and the trigger will become pretty nice and manageable!
O shoot mu Davis @ 21 et and it shoots very high if you use the sights (the front is WAY too short) The best thing you can do is learn to instinct-shoot the little gun. it is afterall a last ditch gun meant to save your lif at very close ranges.
The .38 Special round is far more powerful than the original .41 rimfire was. In fact the .38 Special is comparable to a 2" revolver and is quite deadly at close range! I would consider it for a close-in belleygun as it could deliver a fight stopping blow.
The small framed .22 long rifle is likewise a fight stopper but you will have to learn to face or head shoot your attacked! At best it should be viewed as a knife with a long blade! Body hits are almost ineffective.
If I were to suggest purchas of a Cobra derringer, I would say ti get the large frame in either .22 Mag, .38 Special or 9mm. The large frame just gives more to hang on to, Powerwise, the .22 mag is at the bottom of the stack but due to the availability of good bullets in the factory loads, you can expect better results from body hits. Both the 9mm and 38 Special offer a bewildering array of bullets for all purposes so finding a favorite is possible.
Just remember this is 1860's technology even when loaded with mofern ammo.