Bishop Libby said: "It was wonderful day to have people from across the length and breadth of Derbyshire and joining us from every context that our diocese covers to gather and in worship and prayer and root my ministry and our shared ministry together in worship.
"I want to thank them all for giving their time and prayers on Saturday.
"It was a very precious moment and it is an extraordinary privilege to be asked to serve God by serving the people of Derby and Derbyshire and that is what [the cathedra] represents."
Hundreds of people gathered to witness the service, which also included garlanding her with flowers, presenting her with the pastoral staff - the bishop's crozier - and vesting her in the diocesan robes and mitre.
The doors of Derby Cathedral were closed before the service started and - again following tradition - Bishop Libby banged on the door three times with her crozier to gain access to the building.
At the end of the service, she stood outside to publicly bless the city of Derby, the county of Derbyshire and the Diocese of Derby.
Earlier in the day, Bishop Libby arrived in Derby by train, accompanied by children from Church of England schools in Derby and Derbyshire.
She then walked through the city, greeting people of all faiths and none, on her way to the Cathedral.
Bishop Libby said the walk had been a wonderful experience: "Clearly people had heard that we were coming... It was really good to say hello to people and explain who I was and to have them wish us well and ask for our prayers and tell us something of their own lives and experiences."
For the final part of her walk, she was joined by faith leaders and civic dignitaries, including the Lord-Lieutenant oif Derbyshire, William Tucker, High Sheriff Lord Burlington and the Mayor of Derby, Councillor Frank Harwood.
Bishop Libby arrived outside Derby Cathedral to be greeted by performers from the National Liturgical Dance Network, who treated her to some lively and colourful dancing before themselves going in to the Cathedral to take part in the service.
On Sunday morning, Bishop Libby presided at communion in Derby Cathedral for the first time.
She will spend time during the first weeks of being in post visiting all of the deaneries in the Diocese.