The former Derbyshire opener scored 107 off 191 balls, sharing a third wicket stand of 163 in 39 overs with Joe Clarke who made 66 from 95 balls.
But the innings imploded with eight wickets falling for 62 runs as Fynn Hudson-Prentice claimed career-best figures of 4 for 36 and Ben Aitchison took 3 for 27 to bowl Notts out for 256.
Stuart Broad responded by taking 2 for 22 before Luke Fletcher, 3 for 23 and Dane Paterson, 2 for 28, left Derbyshire reeling on 86 for 8 at the close.
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Haseeb Hameed scored two centuries in the previous game but was out to the second ball he faced, edging Luis Reece to first slip.
When Hudson-Prentice found lift to have Ben Duckett caught at second, Derbyshire’s decision to put Notts in looked good but they squandered the initiative by bowling short and wide.
Slater and Clarke took full advantage, the latter passing 5,000 first-class runs while his partner more than made up for his pair against Derbyshire last season while on loan with Leicestershire.
Clarke pulled Reece for six and repeated the stroke against Sam Conners on his way to a 69 ball fifty, his second of the season.
Slater reached his hundred from 160 balls by clipping Aitchison through midwicket but Clarke’s dismissal, fencing at Aitchison, plunged the innings into decline.
Hudson-Prentice was rewarded for bowling full when he trapped Lyndon James and Steven Mullaney lbw and when Slater was caught in front pushing half forward at Matt Critchley to the second ball after tea, Notts had slumped to 205 for 6.
Tom Moores and Liam Patterson-White counter-attacked briefly but the innings folded quickly after Moores miscued trying to pull Reece.
After the last three wickets fell for two runs, Broad had Reece caught behind before Tom Wood failed to beat Luke Fletcher’s throw from midwicket.
Broad removed Wayne Madsen with late inswing, Leus du Plooy was lbw shuffling across to Paterson and Fletcher trapped Harvey Hosein on the crease before Critchley edged Paterson behind.
Fletcher then sent back George Scrimshaw and Hudson-Prentice to complete a day that promised a batting bonanza but ended in bowling carnage.