Wow ... I must say , a very interesting question !!!
Lets find faults in the very first statement :
x^2 = x+x+x+x+x..................(x times)
This is not valid if x is negative ( How do you define -2 times) Also not valid if x is not an integer .. ( say 4.5 times) This expression is only valid for positive integers... and at no other place.
On the other hand x^2 is a continuous function.
So the equivalence of the LHS and RHS is not valid.... and thus the problem while differentiating...