Leeds made the trip to Pride Park the fourth round of the Carling Cup looking for a place in the quarter finals.
Leeds made the worst possible start when Emanuel Villa scored after only six minutes.
Things got worse with less than twenty minutes on the clock when Nathan Ellington put the hosts 2-0 up.
This sprung Leeds into life and they enjoyed long periods of possession, they eventually got a deserved goal back five minutes before half time when Becchio rose to head past Carroll in the Derby goal.
Leeds didn`t want to hear the half time whistle and finished really strongly.
Derby had a couple of early efforts easily dealt with in the second half and then Leeds once again started to control the game.
Snodgrass was unlucky with a free kick when his shot hit the post with the keeper beaten.
Next chance fell to Andy Robinson whose shot was fantastically saved by Carroll.
Leeds decided to really go for the equaliser and brought on leading goal scorer Jermaine Beckford with twenty five minutes remaining.
Beckford was quickly into the game and had a good effort just wide of the post.
Hulse should have wrapped the game up with five minutes left when a mad flurry in the Leeds area saw his effort come back off the post.
Beckford had Leeds last chance deep into injury time when his effort just went over the bar.
Disappointed that Leeds have gone out of the Carling Cup but a good team performance after a dreadful start to the game.
Once again the centre of defence looks Leeds problem area.
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