I could not find fred or frederick for derbyshire census at all both in 1891 or 1901, I also could not find his registery of birth in the bmd, I found one fred Hagin but I don't think this is correct but here it is,
1891 England Census
about Fred Hagin
Name: Fred Hagin
Age: 8
Estimated Birth Year: abt 1883
Relation: Son
Father's Name: Tom
Mother's Name: Augusta
Gender: Male
Where born: Liverpool, Lancashire, England
Civil Parish: Great Boughton
Ecclesiastical parish: St Paul
County/Island: Cheshire
Country: England
Registration district: Chester
Sub-registration district: Chester Castle
Household Members: Name Age
Augusta Hagin 46
Fred Hagin 8
Maud Hagin 12
Sam Hagin 14
Tom Hagin 56
Tom Hagin 14
Walter Hagin 20
I had found a Hagins in Gossop, but no Fred and I found a benjamin F Hagin born in Chesterfield in Mar 1922, and a Thos. Hagin born in Chesterfield in 1874, but was living in Yorkshire by 1881. Sorry I could not be much help on this one.