26 year old Thirlwell joined the Pride Park club last Friday for a undisclosed fee after being transfer-listed by Neil Warnock this summer.
The Newcastle born Midfielder spent seven years at Sunderland before joining the Blades in July 2004, whilst with the Black Cats he spent a short time at Swindon Town on loan during the 1999/00 season.
During his Bramall Lane career which lasted one season he notched up 28 games (8 as sub) and scored one during the 3-3 home thriller against Wolverhampton Wanderers.
He made his full Rams debut in the 1-1 home draw against Brighton & Hove Albion, former Blades striker Paul Peschisolido netted for the Rams.