Though many of the topics we have covered to this point were explained in an intuitive manner in your previous calculus courses, it is likely that the concept of differentiation was covered in a more rigorous fashion. This is due to the fact that most of the interesting properties and consequences of the derivative are easy to prove though this does not detract from their importance. The reason for this can be attributed to the fact that the derivative is defined as a limit and, even with only a limited understanding of limits, many results about the derivative are direct consequences of the properties of functional limits and continuity. This begs the question of why we should revisit this topic if you have already seen it presented in detail. The first answer to this question is simply that ignoring the theory of differentiation would leave our story with a large plot hole. The derivative has been a key player in the study of analysis for several hundred years and therefore deserves a place of prominence. The second answer is somewhat more satisfying. Yes, we will explore ideas you are familiar with, but our perspective will be of a theoretic nature rather than the computational and application-driven perspectives of calculus. And, we will delve deeper and consider functions with much more bizarre behavior than those you have previously attempted to differentiate.
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