Hameed became increasingly assured as the session wore on, with three fours in a Fynn Hudson-Prentice providing the highlight.
The England opener reached a maiden First-Class fifty for his new county with a cover-driven four off Ben Aitchison.
But by then, wickets had started to tumble with Nash lbw to Connors and Ben Duckett caught low down at second slip to give Aitchison his maiden First-Class wicket.
Derbyshire had declined to use Cohen before lunch, but when he was introduced, he struck in his second over when Joe Clarke edged an attempted drive behind.
Reece, who was Derbyshire’s leading wicket taker in the championship last season, then snared a quickfire double as Hameed was lbw playing forward and, two balls later, Steven Mullaney was well caught in the gully. Peter Trego’s first innings for his new county ended with an edge off Cohen for a single.
The sight of Patel striding to the wicket at number eight was a reminder that Derbyshire still had work to do to wrap up the innings, and the all-rounder engineered a recovery that took his team to respectability and three batting points.
After Tom Moores was caught behind off Hudson-Prentice for 15, Patel dominated an eighth wicket stand alongside Joey Evison with back foot drives and cuts punishing the bowlers when they strayed in direction.
Cohen was dispatched for three consecutive fours, but responded by having Patel caught at slip.
But the momentum had swung again and Jake Ball tucked into Matt Critchley, driving him over the long-off boundary and then swatting him over extra-cover for a second six.
Joey Evison's mature 38 took Nottinghamshire past 300, including another six off Critchley over cover. And although spinner Matthew McKiernan brought an end to the late entertainment in successive overs, the day ended as it began with Notts on top.
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