# What is the derivative of tan^4(3x)?

Apr 9, 2015

If is helpful (until you really get used to it) to remember and rewrite:
${\tan}^{4} \left(3 x\right) = {\left(\tan \left(3 x\right)\right)}^{4}$

This makes it clear that ultimately (at the last), this is a fourth power function. We'll need the power rule and the chain rule. (Some authors call this combination "The General Power Rule".) In fact we'll need the chain rule twice.

$\frac{d}{\mathrm{dx}} \left({\left(\tan \left(3 x\right)\right)}^{4}\right) = 4 {\left(\tan \left(3 x\right)\right)}^{3} \left[\frac{d}{\mathrm{dx}} \left(\tan \left(3 x\right)\right)\right]$

$= 4 {\tan}^{3} \left(3 x\right) \left[{\sec}^{2} \left(3 x\right) \frac{d}{\mathrm{dx}} \left(3 x\right)\right] = 4 {\tan}^{3} \left(3 x\right) \left[3 {\sec}^{2} \left(3 x\right)\right]$

$\frac{d}{\mathrm{dx}} \left({\tan}^{4} \left(3 x\right)\right) = 12 {\tan}^{3} \left(3 x\right) {\sec}^{2} \left(3 x\right)$