The 24-year-old forward initially joined the Rams on a one-month loan deal in February, making his debut as a second half substitute in the side's narrow 2-1 defeat to Watford at Vicarage Road.
Martin went on to make 13 League appearances for the Rams, scoring two goals manager Nigel Clough said: "Chris did really well whilst here on loan so we're pleased to get him signed up, and it's another important signing for us.
"He's a striker with different qualities to what we've already got, especially with his physical presence and ability to hold the ball up.
"We've admired him from afar for some time and monitored his progress closely, and once we got him in the building we knew he would fit in just fine with how we want to play and this group of lads.
"We can't wait to see him in action come the start August with a full pre-season behind him."