Cotton, 22, has played 17 first-class matches since his debut in 2014 and featured regularly in white-ball cricket this season.
Taylor, 21, was ruled out for the season with a back injury.
"They are both exciting fast bowlers who have the ability to win games for Derbyshire," head coach John Sadler told the club website.
"And we must remember they're still only 22 and 21 respectively."
Derbyshire are bottom of Division Two, having failed to win any of their Championship matches this season.
Fast bowlers Ben Cotton and Tom Taylor have signed new contracts with Derbyshire until the end of 2018.