# How do you use the chain rule to differentiate tan(ln(4x))?

Dec 27, 2017

${\sec}^{2} \frac{\ln \left(4 x\right)}{x}$

#### Explanation:

Simply break it down piece by piece:

$y = \tan \left(\ln \left(4 x\right)\right)$

First differentiate the outermost function (the tan function) and apply the chain rule:

$\to \frac{\mathrm{dy}}{\mathrm{dx}} = {\sec}^{2} \left(\ln \left(4 x\right)\right) . \frac{d}{\mathrm{dx}} \ln \left(4 x\right)$

Now differentiate the $\ln \left(4 x\right)$ and keep going:

$\frac{\mathrm{dy}}{\mathrm{dx}} = {\sec}^{2} \left(\ln \left(4 x\right)\right) \frac{1}{4 x} . \frac{d}{\mathrm{dx}} \left(4 x\right)$

$= {\sec}^{2} \left(\ln \left(4 x\right)\right) \frac{1}{4 x} .4 = {\sec}^{2} \frac{\ln \left(4 x\right)}{x}$