A derivative citizenship is one that is derived from someone else’s naturalization or citizenship status. If your immigrant ancestor stated that they were naturalized and they were born outside the country, look closely at the citizenship of the parents–especially the father. Your foreign-born ancestor who claimed in a US census record he was naturalized at the age of five may have been–through his father’s naturalization.
In pre-1906 US naturalization records, children and the spouse are not mentioned, but their citizenship status would have followed the father’s through his naturalization. The exception would be children at the age of majority when the father naturalized.