Why isn't this valid? If I have an equation, say $x^2 + 9x-93=0$, why can't I just take the derivative on both sides to get $2x = -9$ and therefore $x = -\dfrac{9}{2}$. What is the reason behind why what I did is wrong?
Your equation only holds for certain values of $x$. If you have two functions, places where they are equal are not necessarily places where their derivatives are equal.

If you have an identity, though, you can take the derivative of both sides and keep it valid. For instance: $$\frac{d}{dx}(\sin^2(x) + \cos^2(x)) = \frac{d}{dx}1$$

$$2\sin x \cos x - 2 \cos x \sin x = 0$$

$$0 = 0$$
Differentiation needs two variables that have a mutual dependence.  You have only one variable.

Note the more complete way to describe differentiation is differentiation with respect to a variable.

That last bit is what you cannot do here.  $x$ is related to no other variable.

The expression $x^2+9x-93=0$ is equivalent to stating that x has two possible values ( the roots of that expression.

So it's equivalent to saying something like :

$$x=constant$$

Now it's a trap I fell into myself to say that we can differentiate this and get $1=0$, but in fact there is nothing to differentiate with respect to.  All we have is a statement of equality and, again, no dependence.

Why fall into this trap ?  Because we start out with a mindset that we should be differentiating something and because that sexy $x$ looks beguilingly like a variable.  But it's just a no-good cheap constant under that variable make-up.

Be wary of such disguises.