Cotton, 22, has played 17 first-class matches since his debut in 2014 and featured regularly in white-ball cricket this season.
Taylor, 21, was ruled out for the season with a back injury.
"They are both exciting fast bowlers who have the ability to win games for Derbyshire," head coach John Sadler told the club website.
"And we must remember they're still only 22 and 21 respectively."
Derbyshire are bottom of Division Two, having failed to win any of their Championship matches this season.

The concert was held amid tight security at the heavily protected Intercontinental Hotel.
It was a rare music concert in the city.
Kabul has witnessed a series of suicide attacks and explosions in the past few months.
Saturday evening's concert was originally scheduled to be held in Ghazi stadium to mark the country's Independence Day. More than 3,000 tickets were sold at premium rates.
But the authorities said they could not guarantee security at that venue.
Ms Sayeed did not give up. She went ahead with her performance, but at a different venue.
Her fans were delighted.
"Despite the threats, I didn't think such a big number of girls would attend here. Fortunately, the number of women is bigger than men," said Bahar Sohaili, who attended the concert..
"I asked some of the girls: 'Why you are here?' They told me they were here to defy those who were against the concert."
Ms Sayeed, described by her fans as Afghanistan's Kim Kardashian, is known for her long hair and figure-hugging outfits, which are seen as taboo by many in Afghanistan.
Her music mixes traditional and folk songs rendered as pop. She sings both in Dari and Pashto, Afghanistan's two main languages.
A day before the concert, I interviewed Ms Sayeed. She said she was determined to go ahead with the concert, despite the threats.
"I am taking this as a very good excuse to be able to give a bit of happiness to my fellow Afghans, who are in dire need of it at the moment," she said.
She has received a number of death threats from those who say her outfits and public shows are an affront to Afghan culture.
"There are certain people in Afghanistan who have been against music, against celebration, against even New Year, Eid and everything. I feel like today we need to stand together and fight against it," said Ms Sayeed.
"At the end of the day, we are also human beings and these are basic human needs - music, celebration, Independence Day, New Year."
Ms Sayeed said she would donate the proceeds from the concert to the families of those killed by Islamist militants in an attack on the village of Mirza Olong in northern Sar-e-Pul province.

The body will also be responsible for the registration and regulation of youth support workers and work-based learning practitioners.
They join more than 75,000 school and further education teachers and learning support staff on the national register.
The council's chief executive Hayden Llewellyn said regulating all these groups was a "world-first for Wales".
He added it was "a bold and welcome step".
"By bringing non-formal and in-work education in line with formal education, we are enhancing the status of the often overlooked professions of youth work and work-based learning and the increasingly important role they play in young people's wellbeing and education," Mr Llewellyn said.
Keith Towler, chairman of CWVYS (Council for Wales of Voluntary Youth Services), said good quality youth work "saves young lives".
"The trusted relationship between young people and youth workers and youth support workers is vital for the development of resilience, self-esteem and delivery of professional responses to a huge range of issues," Mr Towler said.
"Recognition of informal and non-formal learning in a wider educational context is important and this will support the development of quality youth work provision."
Julie James, minister for skills and science, said the new registration system "supports our ambition to achieve the highest possible professional standards throughout the whole education sector and it will provide a vital component in ensuring the people of Wales receive high quality learning throughout their lives".
Fast bowlers Ben Cotton and Tom Taylor have signed new contracts with Derbyshire until the end of 2018.
[NEXT_CONCEPT]
Hundreds of young men and women attended a concert by Afghan pop star Aryana Sayeed in the capital Kabul despite opposition from conservatives, and threats of an attack.
[NEXT_CONCEPT]
Wales' youth workers will be regulated by the Education Workforce Council for the first time from 1 April.