How do you find the derivative of the function  y = tan^4(3x)?

You can use the Chain Rule deriving the ${\left(\right)}^{4}$ first then the $\tan$ and finally the argument $3 x$:
$y ' = 4 {\tan}^{3} \left(3 x\right) \cdot \left(\frac{1}{{\cos}^{2} \left(3 x\right)}\right) \cdot 3 = 12 {\tan}^{3} \frac{3 x}{\cos} ^ 2 \left(3 x\right)$