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First-half goals from the excellent Wayne Routledge and Mikele Leigertwood were enough to give Rangers their first away win under Portuguese coach Paulo Sousa.
Derby had little to offer after a bright start flattered to deceive and once Rangers had gone in front, the belief and confidence drained out of the home side.
By the end, there were big gaps all around Pride Park as the home supporters headed for the exits contemplating the harsh reality that, although the manager has changed, the team has not.
Clough will have been aware of Derby's shortcoming before he took the job but this performance will have underlined the problems he will have to solve to stop the Rams being dragged into a bitter fight for survival.
The picture looks far brighter for Rangers, who recovered from an uncertain opening to dominate the match and only the reflexes of goalkeeper Roy Carroll saved Derby from further damage in the second half.
Once Routledge had put his team in front, Rangers controlled the play with all of their midfielders enjoying an impressive afternoon.
Derby were a little unfortunate to fall behind in the way they did as Martin Albrechtsen had made an excellent challenge on Heidar Helguson only to see the ball break into the path of Routledge, who drilled a low shot into the net from eight yards.
But the home side could expect no sympathy for the second goal which was a catalogue of defensive errors.
Mo Camara, who had an uncomfortable afternoon against Routledge, was once again beaten by the winger, who rolled a pass into the feet of the unmarked Leigertwood and he placed a shot from 15 yards into the bottom left-hand corner.
Derby did rouse themselves briefly at the start of the second half but Radek Cerny saved well from Miles Addison and then Hogan Ephraim headed off the line.
It needed a smart save from Carroll to deny Ephraim a third and the only disappointing note fro the visitors was an injury to their captain Martin Rowlands, who was stretchered off after only 12 minutes with what is thought to be a serious knee injury.