The mathematician who formalised differential calculus and who is responsible for the notation in use today is Gottfried Leibniz. However, in has calculation of pi, Aristotle used the principle of convergent series and limits more than 2000 years earlier.
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Just about all of calculus is based on differential and integral calculus, including Calculus 1! However, Calculus 1 is more likely to cover differential calculus, with integral calculus soon after. So there really isn't a right answer for this question.
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http://en.wikipedia.org/wiki/History_of_calculus Have a look at this wikipedia article. It has a great history of calculus.
Gottfried Leibniz is called the father of integral calculus.