Taking the Derivative of Both Sides of an Equation If we have an equation like 

y = x^2

This implies that 

y’ = 2x

If we have an equation like 

x = 4x^2

and we take the derivative of both sides we get

1 = 8x

With the solution x = 1/8, which is not the solution to the original equation. This is instead the value where the slopes of both sides of the equation are equal.

So when (and why) is an equation resulting from the derivation of both sides of some original equation implied by the original equation.
Your problem comes from thinking that you take derivatives of equations. But you don't. You take derivatives of functions.

For example, you can think of the equation
$$
x^2 -1 = (x-1)(x+1)
$$
as telling you two different ways to write the same function of $x$. When you differentiate the two expressions using the rules for derivatives you will get the same answer, in two different forms.

The equation in your question,
$$
x = 4x^2 ,
$$
does not say that two functions are equal. It asks for the value of $x$ at which those two functions happen to have the same value. That's just a number. It makes no sense to take the derivative.
The solutions of $f(x)=g(x)$ are the points of intersection of $f$ and $g$; the solutions of $f'(x)=g'(x)$ are the points where $f$ and $g$ have parallel tangents.

In our case  $f(x)=x$ and $g(x)=4x^2$.  The meet for $x\in\{0,1/4\}$.  Now $f'(x)=g'(x)\iff x=1/8$, for which $x$ the functions are parallel (but not meet).