A proof of 2=1 using derivative We know that 
$2^2=2+2$

$3^2=3+3+3$

Similarly

$x^2=x+x+...$ ( upto $x$ times)

Now I want to differentiate both side with respect to $x$

This gives
$2x=1+1+1+.....$ ( upto $x$ times)

$2x=x$

Cancelling $x$ I have 

$2=1$

Which should be false.
Where am I going wrong? Please tell. The proof has to be wrong somewhere.
You assume that $x$ is a natural number, but derivatives don't exist for functions on the set of natural numbers since any neighborhood of a natural number contains infinitely many non-natural numbers.
You assume that $x$ is a natural number, but derivatives don't exist for functions on the set of natural numbers since any neighborhood of a natural number contains infinitely many non-natural numbers.