# Question #923ff

##### 1 Answer
Dec 19, 2017

$\frac{\tan \left(2 x\right) + 2 x {\left(\sec \left(2 x\right)\right)}^{2}}{x \tan \left(2 x\right)}$

#### Explanation:

$\frac{d}{\mathrm{dx}} \left(\ln \left(x \tan \left(2 x\right)\right)\right)$
$= \frac{1}{x \tan \left(2 x\right)} \cdot \frac{d}{\mathrm{dx}} \left(x \tan \left(2 x\right)\right)$ (ln(x) derivative and chain rule)

$= \frac{1}{x \tan \left(2 x\right)} \left[\tan \left(2 x\right) \cdot \frac{d}{\mathrm{dx}} \left(x\right) + x \cdot \frac{d}{\mathrm{dx}} \left(\tan \left(2 x\right)\right)\right]$ (product rule)

$= \frac{1}{x \tan \left(2 x\right)} \left[\tan \left(2 x\right) \cdot 1 + x \cdot {\left(\sec \left(2 x\right)\right)}^{2} \cdot \frac{d}{\mathrm{dx}} \left(2 x\right)\right]$ (tanx derivative)

$= \frac{1}{x \tan \left(2 x\right)} \left[\tan \left(2 x\right) + x {\left(\sec \left(2 x\right)\right)}^{2} \cdot 2\right]$
$= \frac{\tan \left(2 x\right) + 2 x {\left(\sec \left(2 x\right)\right)}^{2}}{x \tan \left(2 x\right)}$

(you can do further simplification if you want)