The first quantifier negation law says that
$\neg \exists xP\left(x\right)=\forall x\neg P\left(x\right)$
And the second says that
$\exists x\neg P\left(x\right)=\neg \forall xP\left(x\right)$
Is this a correct way to derive the second one from the first using the double negation law?
Starting with:
$\neg \exists xP\left(x\right)=\forall x\neg P\left(x\right)$
Negate both sides:
$\neg \neg \exists x\neg P\left(x\right)=\neg \forall x\neg \neg P\left(x\right)$
Apply the double negation law:
$\exists x\neg P\left(x\right)=\neg \forall xP\left(x\right)$
It's nearly right, but when we negate both sides we can't negate inside the quantifiers without justification. We need to get the extra negation on the inside a different way.
I would say, if the first quantifier negation law works for any predicate we can use for $P$, then since $\neg P(x)$ is also a predicate for any predicate $P(x)$, it tells us that we must have $\neg \exists x \neg P(x) \equiv \forall x \neg\neg P(x)$ for any predicate $P.$ And then using your idea of negating both sides will give us $\exists x \neg P(x) \equiv \neg \forall x P(x),$ as we intended to prove.
So you're right that the two are very closely related to each other, differing only by negatives, the execution was just slightly off.