how can be this possible? What is wrong with this.

Question

We can see that

1^2 =1 ; 2^2 =2+2 ; 3^2=3+3+3 ; . . . x^2=x+x+x+..... (x times)

differentiation on both sides gives

2x=1+1+1+....... (x times)

2x=x

What's happening hear.How is this possible.

Assume X be as integer and non-integer ,both cases.

Answer 1

This is because, if $x$ is not a positive integer what does, "$x$ times" mean? Like what is, $\sqrt{2}$ times? What does that even mean?

Answer 2

You're trying to use the linearity property of derivatives: ie, that $\frac{d}{dx}(\sum_{i=1}^n f_i(x)) = \sum_{i=1}^n\frac{d}{dx}f_i(x)$, but how do you write $\underbrace{x + x + \cdots + x}_{x \text{ times}}$ in the form $\sum_{i=1}^n f_i(x)$?

Answer: You can't, so you're proof is invalid.