The Derbyshire's in my family tree come from Runcorn, Lancs and the furthest I have researched back to is 1815. I see that you have some link with the fact that a relation of yours is called Samuel Derbyshire born Runcorn 1816 well I have a relation who was born around the same years 1815 in Runcorn called Thomas. He married Alice Thompson in 1840 and had the following childrem all born in RUncorn
James 1837
Elizabeth 1838
William 1840
David 1841
Joseph 1842
Does this seem to relate to your research