Derby eventually made 311 for four with Hasan Adnan making 72, sharing in a stand of 152 with left-hander Rogers for the third wicket.
Most of the championship had some respite from the rain on Friday.
Veteran Somerset opener Peter Bowler resumed after a day off to hit an unbeaten 187 as his side posted 400 for three declared in their second division match against Essex at Chelmsford.
Bowler piled on 232 for the fourth wicket with captain Michael Burns, who cracked 124 not out himself before declaring when the visitors reached their fifth batting point.
It was Bowler's second century of the season.
Essex then reached 28 without loss in reply before the close.
In the First Division, Phil Weston hit a century as Gloucestershire reached 314 for four against Warwickshire at Edgbaston.
Weston put on 120 for the first wicket with Craig Spearman (77) before going on to score 122 from 213 deliveries with 13 fours and three sixes.
Matt Windows (48) and Tim Hancock (44) fell just short of half-centuries as the visitors made a solid start.