Fallacy in differentiation

The derivative of x2, (x squared) with respect to x, is 2x. However, suppose wewrite x2 (x squared) as the sum of x, x's, and then take the derivative: Let f(x) = x + x + ... x (x times)Then f'(x) = d/dx[x + x + ... x] (x times) = d/dx[x] + d/dx[x] + ... d/dx[x] (x times) = 1 + 1 + ... 1 (x times) = x. This argument appears to show that the derivative of x2,(x squared) with respect to x, is actually x. Where is the fallacy?


Answer:

The fallacy lies in ignoring the fact that the number of x's being added is not constant. Not only is x changing, the number of x's is also changing.