A post modem examination revealed she had been shot on two separate occasions.
Peregrine Falcons are protected by the Wildlife and Countryside Act (1981) and killing them carries a penalty of up to six months imprisonment.
PC David Frost, the local wildlife officer said: "Although the bird was found near to Derwent we are unable to establish the location of the shooting.
"We do believe the incident may have occurred during the week prior to her death however. The bird also had a piece of shot from an earlier incident, perhaps a month earlier, which had largely healed."
The dead bird was discovered on Friday, May 12.
PC Frost added: "I feel sure that there will be universal condemnation from conservation groups, sporting groups and landowners following this incident.
"There is a strong partnership emerging in the area concerned in relation to birds of prey, and that just makes this all the more disappointing.
"I appeal to anyone with information about how the bird came to be shot to come forward and help with our investigation."
A reward of up to £1,000 is being offered by the RSPB for information. Call 0114 296 3481 or Crimestoppers in confidence on 0800 555 111.