It appears that at least one web site concerning the Derby surname confuses that surname with that of Stanley, Earls of Derby (also Smith-Stanley), incorrectly listing several Stanley Earls of Derby as being of the Derby surname. The current Earl of Derby is Edwad Stanley. http://en.wikipedia.org/wiki/Earl_of_Derby
Possibly causing some of the confusion is the fact that Edward George Geoffrey Smith-Stanley, 14th Earl of Derby, who was Prime Minister of the United Kingdom under Queen Victoria, was at times referred to as “Lord Derby” alluding to his title not his surname. Elsewhere he is called Lord Stanley. He probably wasn't the only Stanley Earl of Derby to be called Lord Derby. Also, Derby, British Columbia was named for him although the Stanley Cup is named for his second son, Frederick Arthur Stanley, a Governor-General of Canada. Someone seeing the names Derby and Stanley used interchangeably may have missed the fact that, again, Derby was the title, Stanley was the surname.
This does not completely rule out a possible connection between the two families by marriage, a younger son changing his name, etc. But I checked The Complete Peerage, vol iv, Burke’s Peerage, Burke’s Landed Gentry, vol I, and Burke’s Commoners to see if there was any such connection but I wasn’t able to find anything.