The error actually occurs at the beginning:

For x > 0,

x = 1 + 1 + 1 + 1 .... (x times)

That's just not true in general. It's only true for integers (like -5, 0, 1, 10). For example, what does the expression mean if you set x = 0.5? Or even worse, what about x = pi?

Why doesn't the proof work if x is an integer? That would still "prove" that 2 = 1.

The rule,

D(x^2) = 2x

...is not a rule for a particular number x. It is a rule for the function x^2, which is a continuous (and hence differentiable) function.

What the proof is trying to say is something like this:

Hey Monqui, you know that 2x is the derivative of x^2, right? Now that's true at every point x, so in general the derivative of a real number must be twice its square root, and in particular this must be true of an integer.

But that's certainly not true. Let's take k = 16. Then D(k) = 8, right? But now express k as part of the function 2x. D(2x) = 2 for all x, so D(k) = D(8) = 2. But then 2 = 8.

The point is that there is no such thing as the derivative of a number. Only functions have derivatives.