Naylor, formerly of Wolves and Celtic, has been training with the Rams and played in their last two under-21 fixtures. He was a free agent after his contract with Sky Bet League Two club Accrington was cancelled by mutual consent.
The 33-year-old will provide cover for regular left-back Craig Forsyth during the closing weeks of the campaign.
Naylor began his career at Wolves where he worked under academy manager Chris Evans, who is now head of football operations at Derby.
He played almost 300 league games for Wanderers before leaving to join Celtic in 2006.
After four seasons in Scotland a move to Cardiff followed, before signing for Accrington last summer.
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- Craig Forsyth