Gilchrist was not a common name in Co Derry. In the 1831 census there were just 3 households, all in the same parish – Tamlaghtard. 1 in the townland of Upper Doaghs and 2 in Ballymaclary. All were Roman Catholic.
Death registration started in Ireland in 1864. I searched the records 1864 to 1901 for Patrick & Frances but without success.
Tamlaghtard RC only has baptism & marriage records from 1863 onwards so if that’s where your family came from then sadly there are no records to search.
No Patrick Gilchrist in the Tamlaghtard area in Griffiths Valuation (1858), nor in Co Derry. How sure are you that your family came from Co Derry?