The bowling all-rounder joined Derby at the start of the county championship season from Surrey, but has struggled for form since making the move and was dropped from the team in recent weeks.
The decision to relinquish the captaincy was made following discussions between Clarke and head of cricket John Morris following the recent LV County Championship match against Worcestershire at New Road.
Morris said: "The decision has been made after several discussions between Rikki and myself. He felt as though the role was impacting on his own personal performance and that was affecting the team."
Clarke, who played two Tests and 20 one-day internationals for England, has scored just 296 championship runs at an average of 21 and taken 11 wickets at 38 apiece this season.
Australian Chris Rogers, Derbyshire"s overseas player, will take over the captaincy until the end of the season.