A number of routine operations in the Royal Derby Hospital have been suspended to keep beds clear for cold weather injuries.
It has postponed the treatment or surgery of 51 patients because beds are being used for emergencies.
More snow overnight has led the emergency services to warn they will take longer to respond and about 200 schools are closed.
The council said it was concentrating on gritting main roads only.
As well as grit supply problems, council officials have said the low temperatures mean gritted roads need to be treated more often.
A number of roads have been closed, mainly in the Peak District, and train services reduced.
The police have warned frost has opened up potholes in some roads, leading to an additional driving hazard.