Davis, 20, has made three Championship appearances this season, with his 7-146 against Glamorgan the best 2016 figures by a Derbyshire bowler.
Hemmings, also 20, made his first-class debut against Worcestershire in June.
"They will benefit greatly from another winter working with the first team," head coach John Sadler said.
Derbyshire pace bowler Will Davis and all-rounder Rob Hemmings have signed two-year contracts, keeping them with the county until 2018.