The home side were put under pressure by fine bowling from Ravi Rampaul, who claimed figures of 4-56 for Derbyshire, producing impressive spells with the new ball in hand.
However, Lees and Burnham played the patient knocks that were demanded by head coach James Franklin in the aftermath of the club’s defeat to Gloucestershire last time out. Although neither batsmen was able to make three figures, Durham ended the first day in a decent position at 254-8.
Durham were inserted after Derbyshire captain Billy Godleman opted for an uncontested toss. Michael Jones became the fourth player to open alongside Lees for the home side in the Championship this season. However, Jones lasted just four overs when he edged a Rampaul delivery behind to Harvey Hosein for a duck.
Cameron Bancroft and Lees attempted to see off the new ball. The two players were disciplined in their efforts, but Rampaul was able to remove Bancroft off the penultimate ball of his eighth over, trapping the Durham skipper lbw for 18. Lees survived a run out attempt when he was stranded in the middle of the wicket as Tom Lace missed the stumps.
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He made the most of his opportunity, putting on a partnership of 57 with Gareth Harte for the third wicket. They took Durham into the lunch break, but their stand was broken when Harte missed a straight one from Tony Palladino to fall lbw for 24.
Lees kept the home side on course and was rewarded for his patience, reaching his half-century from 102 deliveries. He built another solid partnership with Burnham, but then the opener played a loose drive and was caught a second slip from a Luis Reece delivery to fall for 63.
Burnham took centre stage, building on the work of his team-mate. He picked his moments when to push for boundaries, forming another impressive stand for the hosts with Liam Trevaskis. The duo were able to reach their stand of fifty and push Durham beyond the 200-run mark, earning the first batting point of the contest.
Burnham reached his second Championship fifty of the season from 129 balls, including three fours and a six over mid-wicket. Trevaskis seemed on course to join his team-mate, only to lose his wicket for 40 playing a poor stroke to a short ball from the leg-spin of Matt Critchley.
The timing of the wicket could not have been worse for the home side as Rampaul returned with the new ball. The paceman did the damage once again, causing a collapse in the Durham ranks. He removed Burnham for a fine innings of 67 lbw, and he produced a perfect ball to remove Ben Raine for a golden duck.
Brydon Carse survived the hat-trick ball from the West Indian and added 10 runs to take his team past the 250 and a second batting point. Ned Eckersley and Matty Potts remained unbeaten at the close, leaving the hosts in a solid position heading into day two.