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.
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.