Curiouser and Curiouser

         Continuing my work on paradoxes, I did some research and found a paradox that I couldn't explain.  From this I obtained a general form for the paradox in which, when multiplication is reduced to a series of additions, the process of differentiation breaks down and we obtain two different answers.  To elaborate, I provide the general form for any polynomial in x here:

          C.xⁿ + ... = C. x^{n - 1}(a + a + ... + r) + ...  (where r is the remainder of x / a)  ---(1)

          d(C.xⁿ + ...) / dx = C.nx^{n - 1} + ...                                                                                ---(2)

            d(C.xⁿ + ...) / dx = C(n - 1) x^{n - 2}(a + a + ... + r) + C.x^{n - 1}(0) + ...    (from (1))

                                            = C(n - 1)x^{n - 2}(x) + 0 + ...

                                            = C(n - 1)x^{n - 1} + ...                                                                           ---(3)

                                         n = n - 1                                                (from (2) & (3))     ---(4)

          The logic behind this is that xⁿ is a shorthand expression for x.x.x, x times, which is shorthand for x + x + ..., x times x times x times ... and lastly, any number x is shorthand for a + a + a + ... + r (where r is the remainder when x is divided by a), e.g. We could say the number '23' is short hand for 4 + 4 + 4 + 4 + 4 + 3.

        As an illustration, the original paradox was given as:

                            x² = x + x + x + ...    (x times)

                   dx² / dx = 2x

                   dx² / dx = 1 + 1 + 1 + ...

                                 = x

                            2x = x      (dividing by x)

                              2 = 1

      Here C = 1, n = 2, a = 1, and r = 0.  The final expression is:

           1.2x^{2 - 1} = 1(2 - 1)x^{2 - 1} 

                  2x = x, and so on

          OR from (4)

                    2 = 2 - 1

                    2 = 1