Paul Green put the Rams ahead at Saltergate but second-half goals from Scott Bowden and Robert Page gave the Spireites the win.
The Rams were without Rob Hulse, Dean Leacock, Kris Commons, Steve Davies and Chris Porter, while Green also went off injured but Clough was upbeat.
He told the club's official website: "Overall we were very good first half and a little sloppy in the second. Some of the football we played was very pleasing.
"I thought the players did very well during that first period, created some excellent chances and probably should have had more than one goal.
"The mistakes we made during the second half cost us but I'd sooner have us make those mistakes now and learn from them, than happen when the Championship season gets underway."