Derby claimed a dramatic 1-0 victory but the post-match talk was dominated by shocking chants about Doughty, who was found dead at his home at the age of 54 last month. The Rams will issue a formal apology to their nearest rivals after chants of "where's your chairman gone?" and "you're going down with your chairman" from a small section of the home support.
Forest manager Steve Cotterill condemned the chants after the match, while his counterpart at Derby Nigel Clough also made his displeasure clear.
Derby have responded by issuing an apology and launching an investigation into the matter.
A club statement read: "Derby County Football Club wishes to formally apologise to the family of Nigel Doughty, Nottingham Forest Football Club and all supporters in attendance at Pride Park Stadium on Tuesday night for the behaviour of a small group of people at last night's game.
"Some of the chants directed to opposing supporters were wholly inappropriate and damage the excellent reputation of the Club and its supporters at large."
Derby chief executive Tom Glick, who had previously made an apology to Forest club officials, added on the club's official website: "We are very disappointed that a brief moment of insensitive and distasteful chanting by a minority of supporters spoiled what was a great occasion at Pride Park Stadium last night and we do not condone the actions of those individuals in any way, shape or form.
"The vast majority of our supporters come to Pride Park to enjoy watching and cheering the Rams on to victory and generate a positive atmosphere inside the stadium.
"It is a shame for this majority that we today have to focus on a small group of people whose actions reflect badly on us all."
The club has appealed to any fans with information about those who may have been involved to contact the club.
After the match, which was settled by Jake Buxton's strike in stoppage time, Cotterill said: "(Derby boss) Nigel (Clough) will probably be embarrassed by that, I would imagine. Nigel is a good man, I get on very well with Nigel. I'm sure he would be very disappointed with that. And most people who have chanted that, they need to be ashamed of themselves."
He added: "What have they done coming out to a football match and come out with that about Nigel Doughty? They need to be ashamed of themselves. They've used football as a carriage to open it up.
"Unfortunately what you can't do and what Nigel can't do, is he can't stop those thousands doing it.
"God bless his soul Nigel Doughty, he was top man for this football club, one of the best chairman that has probably ever been in the Championship. Not just because of the way he funded it, but because Nottingham Forest was his life as well.
"The worst part about it was that we haven't got a result, but he would have been pretty pleased with our effort and commitment."