One of the spelling variations of Carruthers is Carrothers. The earliest record, that is reliably accepted, is with an Archibald Carrothers, born in 1692 in Dumbarton, Scotland. His family moved to Northern Ireland in 1694. Many of his descendants stayed in N. Ireland, but some of the family moved on to America. The majority of those going to America settled in Canada.
An excellent source for this part of the Carrothers line can be found at this link.