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this post was submitted on 06 Jun 2025
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Read the other replies but this is what clicked it for me:
Between step 2 and 3, you applied the derivative to all of the x's in the sum (x+x+x...) but ignored the x in the "x times".
This nonstandard notation helps to hide that. If you wrote this in sigma notation, you'd have:
If you differentiate this with respect to x, you can't ignore the x in the sigma limit. When differentiating a summation where the limits are a function of the target variable I believe you need to use Leibniz rule(?), but I'll leave it there
You cannot differentiate a sum when the variable being differentiated is used to define the number of terms in the sum — unless you rewrite the sum as a closed-form, continuous expression. Even in Sigma notation as you used.
The act of summing “x terms” as you expressed in your sum is not itself a differentiable process.
Once you turn it into a continuous function (in this case x^2), then you can differentiate it.
The Leibnitz rule doesn’t do anything here because you still have an unextractable “x” that’s defining your summation.
That makes sense, thanks. I knew someone would know better than me how to interpret that.