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.
Cotterill paid tribute to Doughty, and 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."
Clough said the Rams would investigate the chants and issue a apology to Forest.
He said: "The club will do that (apologise), we'll do that officially with the club and Tom Glick (Derby chief executive). We'll have a chat about that later and get something out officially."
Asked about an investigation being launched, he added: "I'm sure we will look at it in the next few days. I'm the manager of the football, that's more of a club issue. I was aware of a couple of chants, but you get so engrossed in the game."