$$ y = x^2 $$ $$ y = \underbrace{x+ x+\cdots +x}_{x\text{ times}}$$ $$\frac{dy}{dx} =\underbrace{1+ 1+\cdots +1}_{x\text{ times}} $$ $$\frac{dy}{dx} = x$$
But the real answer is obviously $2x$. I know that I am making some kind of huge mistake but just can't find out where.
The derivative of $x^2$ is not the derivative of the expression $\displaystyle\sum_{n=1}^x x$, because the number of terms of the expression also depend on $x$. Let's look at your logic. You are saying that: $$ \frac{d}{dx} (x^2) = \frac{d}{dx} (\displaystyle\sum_{n=1}^x x) = \displaystyle\sum_{n=1}^x \frac{d}{dx} x = \displaystyle\sum_{n=1}^x 1 = x $$ Which is wrong because the sum depends on $x$. Taking it out would be treating it like a constant, which it is not, because it depends on $x$.
What if $x=\frac12$? Your representation fails.
In addition, you have completely disregarded the variable as an upper limit of the sum when differentiating.
So your formula is incorrect.
There may be a better formula that takes these things into account.
