Is this solution correct? Prove whether $f(n)$ is $O$, $o$, $\Omega$, $\omega$ or $\Theta$ of $g(n)$

Question

$f(n) = n + (\log n)^{2}$, $g(n) = n + \log(n^{2})$.

log is assumed to be base 2

So I differentiated each of the functions

$f(n)' = 1 + 2(\log n)$

$g(n)' = 1 + (2/n\ln(2))$

and then finding the limit of $f(n)' / g(n)'$ as $n$ approaches infinity. Which the denominator approaches 1, while the numerator approaches infinity. Thus, $f(n) = \omega g(n)$?

Is this correct? I'm pretty new so if you could guide me I'd appreciate it!