What is wrong in this proof?
The proof states that $(\underbrace{x+x+\dots+x}_{x~ times})' = (\underbrace{1+1+\dots+1}_{x~ times})$. Let's give it a try.

$$\lim _{\epsilon \to 0} \frac{\underbrace{x+\epsilon+x+\epsilon+\dots+x+\epsilon}_{x+\epsilon~ times} - \underbrace{x+x+\dots+x}_{x~ times}} {\epsilon} =\\ \lim _{\epsilon \to 0}\frac{ \underbrace{\epsilon+\epsilon+\dots+\epsilon}_{x~ times} + \underbrace{x+\epsilon+x+\epsilon+\dots+x+\epsilon}_{\epsilon~ times}}{\epsilon} = \\ \lim _{\epsilon \to 0}\frac{ \underbrace{\epsilon+\epsilon+\dots+\epsilon}_{x~ times} + \underbrace{x+x++\dots+x}_{\epsilon~ times} + \underbrace{\epsilon+\epsilon+\dots+\epsilon}_{\epsilon~ times}}{\epsilon} =\\
\lim _{\epsilon \to 0}\underbrace{1+1+\dots+1}_{x~ times} + \underbrace{x}_{1 ~time} + \epsilon = 2x$$