Derbyshire struggled on the second morning under overcast conditions, as Rushworth claimed six-wickets to rip through the visitors’ line-up and hand the hosts a convincing 140-run first innings lead.
David Wainwright top-scored with 18, while Tim Groenewald added 17 and Mark Footitt chipped in with a brisk 16 to help the visitors past the follow-on target before being dismissed for 113.
Borthwick - who was dismissed by Dan Redfern for 117 - and Keaton Jennings (65 not out) produced an outstanding partnership worth 168 runs to hand Durham a commanding 348-run lead at stumps.
Richard Johnson (6) was the first man to go off the fourth delivery of the day from Rushworth, as Derbyshire lost regular wickets to slip from 19-2 to 69-8 in the morning session.
Wayne Madsen (13), Shivnarine Chanderpaul (14) and Wes Durston (16) all made starts but failed to kick on, as the Derbyshire batsman were unable to combat against the swinging ball. Rushworth was the main destroyer for the hosts, finishing with impressive figures of 6-64 from his 14.4 overs.
The Derbyshire tail, however, produced some stubborn resistance – and not for the first time this season. Wainwright (18 not out) and Groenewald (17) added 25 runs for the ninth wicket, before the former added 19 with Footitt – who hit 16 from just five balls – as the visitors recovered slightly to be bowled out for 113, trailing Durham by 140 runs.
Despite claiming the early wicket of Mark Stoneman (10), the hosts made an excellent start to their reply courtesy of Borthwick and Jennings. Together, the pair added 104 runs before the tea interval to move Durham onto 126-1 at the break, a lead of 266 runs.
The duo continued their flourishing partnership which pushed the lead past 300 runs midway through the evening session. Borthwick duly completed his century from just 117 balls, including 16 fours and one six, before partner compiled his half century from 136 balls.
The superb 168-run stand was finally broken by Redfern in the 54th over, as Borthwick was dismissed for 117 off 155 balls to leave Durham 190-2, leading by 330 runs.
The hosts added another 18 runs before the close, with Jennings remaining unbeaten on 65 from 175 balls and Will Smith 13 not out, as Durham ended the day in control on 208-2.
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