The 38-year-old has signed until the end of the season with the Sky Bet Championship club, who are managed by his former Chelsea and England team-mate Lampard.
Cole, who earned 107 caps for his country, was a free agent after his recent departure from Major League Soccer franchise Los Angeles Galaxy.
Cole has vowed to fight to help Derby reach the Premier League, with the Pride Park club currently sixth in the Championship table.
“This is probably the last interview where I call him Frank,” said Cole.
“I respect him so much, when I played with him and what he’s done in the little time he’s been here. He’s my boss now, it’s not as though I’ve come here to take shortcuts or expect anything more because he’s my friend. No.
“The same with Jody Morris and the other coaches. I respect them, they are my bosses, and I’ve just got to get my head down, train hard and work hard. I probably wouldn’t have come back to too many other teams. I would say it’s down to Lamps that I’m here.
“But am I looking forward to it? Of course. It’s a good challenge, a hard challenge, but I’ve always enjoyed a challenge.
Cole has made 682 appearances across stints with Arsenal, Crystal Palace, Chelsea, Roma and LA Galaxy. After winning three Premier League titles and seven FA Cups, alongside the Champions League, Europa League and League Cup, Cole is targeting a first promotion of his career.
“I had a conversation with Frank maybe a month, two months ago,” Cole told Derby’s club website.
“I’ve kept in touch with Lamps, and have followed how the team has been doing since Lamps got here. So for me it was no-brainer to come here, get involved in this great environment, and hopefully push for promotion.
“For sure I’m again coming out of my comfort zone. I could have stayed in the MLS and relaxed, but I’ve come here and I want to fight for something, a good cause and try to make promotion. That would be great for the club, for Lamps, and for me as well.”
Lampard feels his side’s promotion push will be boosted by Cole’s arrival.
“I’m delighted that we’ve been able to secure Ashley’s services for the rest of the season,” Lampard said.
“He’s a fantastic player, a quality left-back, and he also brings great experience to the dressing room. I know him very well from our time together at Chelsea and with England, and I also know what he can bring to the group and the influence he will have.
“Ashley is in good shape, which doesn’t surprise me because he’s an excellent professional, and he will compete for a place in the side like every other player here.
“I’ve made my feelings known over the last couple of days just how highly I think of Ashley and I have no doubt whatsoever he will be a positive addition to our squad.”
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