Dee,
Births, deaths & marriages in what is now Northern Ireland, from 1922 onwards, are only on the GRONI website. You should be able to find both 1951 deaths in Derry on that site. You will need to open an account and buy some credits. It costs £2.50 (sterling) to a view a certificate:
https://geni.nidirect.gov.uk
That site is currently down for planned maintenance till 8.00am on Monday morning UK/Irish time.
Here are 2 probate abstracts from the PRONI website which look to relate to your family:
Hegarty William of Culmore Road Londonderry retired milkman died 30 March 1951 at the City and County Hospital Londonderry Administration Londonderry 24 September to John Hegarty café proprietor. Effects £783 4s. 10d.
Hegarty Ellen of Culmore Road Londonderry widow died 26 April 1951 Administration Londonderry 24 September to John Hegarty café proprietor. Effects £304 16s. 8d.
The full probate files should be in PRONI in Belfast. They will copy them for a fee or if you go in person you can view them free. Both died intestate (so neither left a will) but the files should still give some information on their assets and next of kin.