$\displaystyle y = \sqrt{5x^6-12}$
$\displaystyle y = (5x^6-12)^{1/2}$
$\displaystyle y ' = \frac{1}{2}(5x^6-12)^{-1/2}*30x^5$
$\displaystyle y ' = 30x^5\Big(\frac{1}{2\sqrt{5x^6-12}}\Big)$
Am I doing this right?
Yes. You can also check your solutions with this website: Step-by-Step Derivatives