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.
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.
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?
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.