It was the visitors who came closest in the early stages, with Jamie Ward dragging an effort wide after latching onto a misplaced pass.
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Both sides had their share of the attacking possession in the first period, with Reading almost taking the lead through Royston Drenthe in the 18th minute, the former Everton man narrowly missing the target following a swift counter-attack.
Bryson came close again just before the half-time interval, missing the target when left unmarked after Reading had failed to deal with a corner.
The Royals continued to ask all of the questions, squandering the best chance of the match on 65 minutes when Adam Le Fondre missed an open goal following good work by McCleary.
Patrick Banford looked odds on to break the deadlock for Derby moments later when he went one-on-one with McCarthy, but the Royals keeper pulled off a fine save to keep the score level.
Le Fondre did put the ball in the net before the full-time whistle, but he used his hand and received only a yellow card for his efforts.