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.