It was first discovered simultaneously by Isaac newton and another mathematician named Gottfried Leibniz. (see related link). However, the notation that we use today is for the most part that of Leibniz. As far as development, it is still an actively researched field of mathematics with thousands of contributors.
Also, integral calculus is often attributed to Archimedes of Syracuse (c. 250 BC), having been largely ignored after him.
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Leonhard ''Euler''
Calculus was invented independently by Isaac Newton and Gottfried Leibnitz. Newton did it first and showed the actual usefulness of the discipline by deriving the Law of gravity from astronomical data. Leibnitz however developed a much superior notation and terminology, thus it is actually Leibnitz's calculus that we use today.
Calculus; by a long shot.
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.