# How do you differentiate f(x)=3(tan4x)^(1/2)  using the chain rule?

$6 \cdot \frac{{\sec}^{2} \left(4 x\right)}{{\tan}^{\frac{1}{2}} \left(4 x\right)}$

#### Explanation:

$\tan \left(4 x\right) = t$
$\frac{d}{\mathrm{dt}} \left(3 \left({t}^{\frac{1}{2}}\right)\right) = \left(\frac{3}{2}\right) \cdot \left({t}^{- \frac{1}{2}}\right)$
$\frac{d}{\mathrm{dx}} \left(t\right) = 4 \cdot {\sec}^{2} \left(4 x\right)$
combining them and converting t back we get
$6 \cdot \frac{{\sec}^{2} \left(4 x\right)}{{\tan}^{\frac{1}{2}} \left(4 x\right)}$