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Calculus in itself is not hard, it is usually remembering the algebra and previous math classes that is hard. New concepts are introduced in Calculus, but isn't it the same with any new subject? For example, many problems in integration, the actual calculus is not the hard part, it is using all of the algebra and other concepts you have used your whole life to simplify the problem so it is easy to solve.
The purpose of calculus is to solve physics problems.
Analysis is a broader term for Calculus and the theorems behind it. It is studied both with real and complex numbers as real and complex analysis. Usually calculus just deals with the basic problems of differential calculus and integral calculus.
First, you need to learn how to do calculus. This can be accomplished through either taking a calculus class or figuring it out on your own. Next, you apply what you have learned to the problem, eventually arriving at the answer.
Determining the "hardest" calculus problem is subjective and can vary depending on individual strengths and weaknesses. However, some commonly challenging calculus problems involve intricate applications of multiple calculus concepts, such as optimization, related rates, or advanced integration techniques. Problems that require a deep understanding of calculus principles, creativity in problem-solving, and the ability to apply various strategies tend to be considered the most difficult.