Given 'x' not equal to 0, let us consider the follwoing relation:
x + x + x + .... +x (added 'x' times) = x²
Differentiating both sides with respect to x, we get:
1 + 1 + 1 + 1 + .... + 1 ('x' times) = 2x
(Since the derivative of x² with respect to 'x' is 2x).
So we now have:
x = 2x
Cancelling 'x' from both sides, we have:
1 = 2
Now the very obvious question follows:
Where is the flaw ?
The number of times that x's are added is a variable. Had the number of times been a constant, then the method would have worked.
For example, let us consider g(x) = (x+x+....+x)(m times), where m is a constant.
Then, g'(x) = (1+1+....+1)(m times) = m
This is indeed true as d/dx (mx) = m
However in the present case, the number of times is a variable so that the number of times (x) also needs to be differentiated w.r.t x.
Thus, taking f(x) = (x+x+ ....+x) (x times), by means of product rule, we obtain:
f'(x) = (1+1+ ....+1) (x times) + (x+x+...+x) (1 time)
= 1*x + x*1 = 2x.
This is indeed true, since d/dx (x^2) = 2x.