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The Rams began the game on the front foot with Jordan Ibe and Will Hughes causing the Reading defence plenty of problems when taking control of possession, but their best chance of the opening 30 minutes fell to Leon Best, who had a one-on-one effort saved by Mikkel Andersen.
Moments later, Best headed over the crossbar from close range, but the Rams were fortunate not to concede when a mistake let in Hal Robson-Kanu, but Rams stopper Kelle Roos was on hand to keep his strike out.
The home side ended the first half on top, but they wasted yet another opportunity when defender Ryan Shotton missed the target with a header from eight yards.
The second half began as the first ended with Derby in control, and it took a finger-tip intervention from Andersen to prevent Ibe's 18-yard strike finding the top corner.
Derby finally took the lead with 23 minutes remaining after Russell received the ball off Chris Martin before driving past two players and finishing powerfully past Andersen.
The goal motivated Derby to push on for a second and it came with eight minutes remaining when Alex Pearce put through his own net after Martin's cross had caused confusion in the Reading defence.
That all but assured the win for Steve McClaren's outfit and they eased through the remaining action to book their place in the last 16.