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The way you've written x^2, you have defined x to be a positive integer, in fact:
x^2 = \sum_{n=1}^x (x).
Thus, x^2 is not, according to this definition, continuous. When you are differentiating, you are in fact differentiating the x that is summed over, but you're not touching the x in the index, thus the answers don't match up.
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