Differential Calculus is to take the derivative of the function. It is important as it can be applied and supports other branches of science. For ex, If you have a velocity function, you can get its acceleration function by taking its derivative, same relationship as well with area and volume formulas.
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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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xx + sincos
http://en.wikipedia.org/wiki/History_of_calculus Have a look at this wikipedia article. It has a great history of calculus.
Calc 2, then Calc 3, then usually Differential Equations