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Both derivatives and integrals - two of the most important concepts in calculus - are defined in terms of limits; specifically, what happens when something gets smaller and smaller.
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.
It is certainly used in calculus, just as calculus can be used in trigonometry.
Calculus is the correct spelling.