d/dx x2 is clearly 2x
but x2 can be written as x+x+x+x+x+....(x times)
now if u differenciate this u will get 1+1+1+1+1+1+...(x times)
which is x
therefore d/dx x2 is x
so finally you are saying that 2x = x.
take x to LHS,
which gives you,
x=0,
hence the given equation is possible only for x=0. and no other x.
Puneeth you have asked a very intelligent question but unfortunately you have forgotten the fundamentals of derivatives.
Whenever you say y=x2=x+x+x+.......(x times) you are treating x to be a natural no.(1,2,3, etc.) as you can't say 0.1+0.1+0.1+........(0.1 times).
Thus as soon as you write x2=x+x+x+.......(x times) x belongs to N and so y is a discrete function in x i.e it can take only discrete values and derivatives are defined for continuous and not dis-continuous(disctrete) functions.Here you have made a blunder.
I hope I cleared your doubt.
Bye.