Hi There,
I am not sure about any free fishing in Derbyshire (I am sure some other forum members might be able to help you better than me)
If you can get to Sheffield/South Yorkshire there is lots of free fishing for Grayling available on the River Don, River Loxley & Rivelin.
The River Don especially is stuffed with Grayling at the minute with large catches being reported.
I know there are members on here who fish there regularly and could probably provide further information but you could start by looking at SPRITE here:
Sprite
Good Luck.