Why we can't differentiate both sides of a polynomial equation? Suppose we had the equation below and we are going to differentiate it both sides:
\begin{align}
&2x^2-x=1\\
&4x-1=0\\
&4=0
\end{align}
This problem doesn't seems to happens with other equation like $\ln x =1$ or $\sin x = 0$, we can keep differentiating these two without getting "$4=0$", for example. This why I asked about polynomials.
PS: I'm not trying to solve any of these equations by differentiating then. But differentiation or integration helps and solving equations?
I remember that sometimes to solve trigonometry equtions like $\sin x = \cos x$ we had to square both side so we could use the identity $\sin^2x + \cos^2x =1$. Even thought squaring appears to make it worse because we have a new root.

It's important to remember that we can only differentiate functions. When you write the expression
$$
2x^2-x=1
$$
you are no longer dealing with a function. Instead, this expression describes only the solutions $x$ to a given equation. For instance, 
$$
f(x) = 2x-x^2
$$
is a function, but $2x-x^2 = 0$ is not.
