United's inconsistent form this season has seen them fall out of the running for the Premier League title, although they have made it to the last eight of the UEFA Champions League.
A 3-0 defeat to rivals Liverpool at Old Trafford earlier this month represented the latest setback for David Moyes' seventh-placed side.
Yet Carrick has outlined a desire to make amends with victory when neighbours Manchester City, who remain firmly in the hunt for top spot, visit on Tuesday.
"It was a bad day against Liverpool," the 32-year-old told United's official website.
"We suffered just as much as the fans. It's horrible after a game like that. We're well aware of what those games mean to everyone and, believe me, it means the same to the players.
"I've been part of this club for a long time now and you know what means a lot. This is certainly one of those matches.
"It's a one-off game against your local rivals and in a derby you always want to come out on top.
"It's not going to repair any of the damage that's been done in recent months… but we understand the fans and the result they want, and for us as players it's exactly the same."
