Load it with .45-70(hardcast, thunderhead, or brass CFA) and .410(dragon's breath, flechette, or bolo), you will solve the problem one way or the other.
So, a derringer is a small pistol, usually with two shots, made for close-range self-defense. Normally they use, well, pistol rounds, like a 9mm, which has like, 700 joules of energy or someshit like that. When you fire a lightweight gun, you definitely get some kickback from it, even with a pistol round.
A 45-70 is a big-ass rifle round with something like 4000 joules of energy behind it. You uh, you put that in a little derringer and pull the trigger, both you and your target are gonna feel it.
For the comment, hardcast is a hardened bullet alloy (lead bullets are heavy but soft; alloying it with other metals makes it harder) for when you need to get through something thick, like a boar, or a human being's entire fucking body. Not sure about thunderhead or CFA. 410 is a shotgun caliber. Dragon's Breath is a fun little round that replaces the bullet with pyrotechnics, while flechettes spray you with these ugly little things.
A bolo round is apparently a bolo but fired from a gun. I've never heard of it before today but let me just say that I love the weird things people come up with.
I've got that version that shoots 410. Because of the rifling on it comes out in a pinwheel. If you shot me at 10-15", I doubt a single pellet would land.