How do I get Derkeethus out of the cave?

Its a known glitch that he may become stuck. If you are on the PC, then you can use the console to free him:

A workaround for this problem is to use the console and type: 'setstage 76b4a 200' This will complete the quest. Then type 'player.placeatme 0001403E' while in Darkwater Crossing, which will spawn Derkeethus. This will leave the original (and bugged) Derkeethus still in the cave, and it is recommended to leave this NPC alone to avoid further glitches.

Otherwise, you are out of luck.

A better workaround is to move to the location where Derkeethus wants to get to, this is the outside the pass at the East exit, then open the console and type:

PRID 1b08d  
MoveTo Player

Press enter after each line. This moves the original Derkeethus to the correct location for the quest to autocomplete.