Smith will spend the season at Pride Park.
Rams boss Nigel Clough told the club's official website: "We're extremely pleased to bring Adam in on a season-long loan deal.
"He's a young player with a lot of potential and we've been very impressed with what we've seen of him over the last few years.
"Whenever we've seen him on international duty he's stood out as a very solid right-back, and someone that likes to get forward as well.
"As soon as we were aware John was going to leave last week we identified Adam as someone we would like to bring in, so to have the deal done within a matter of days is very pleasing and we're sure he'll be a good player for us.
"It also says a lot that Tottenham are prepared to let him spend the season with us.
"They're obviously aware of our reputation when it comes to young players and there's no doubt it'll be a great experience for Adam coming here and playing a full season in the Championship."