The first one does look to be old.
The shape suggests, especially the ear on the hammer, suggests 1830-1850 Belgium to me, but I don't have anything to back that up.
There hopefully would be some sort of maker's mark on the lockwork, or a proof somewhere on the barrel.
The screw in the lockplate may be correct, instead of a cobbed repair. I've seen that sort of thing before on less expensive guns.
If the other gun says Derringer, it's either a new or old copy of a Henry Deringer (note the one R in his name) pistol.
But, to tell you the truth, the little one looks brandspanking new to me.
It looks like it wants to be a copy of the authentic Deringer, but doesn't have the style or proportions.
Lack of the front sight and machine turned look of the barrel also lead me to believe that it's a modern repop.