# What is the derivative of f(x)=ln(tan(x)) ?

Aug 12, 2014

$f ' \left(x\right) = 2 \left(\cos e c 2 x\right)$

Solution

$f \left(x\right) = \ln \left(\tan \left(x\right)\right)$

let's begin with general example, suppose we have

$y = f \left(g \left(x\right)\right)$

then, Using Chain Rule,

$y ' = f ' \left(g \left(x\right)\right) \cdot g ' \left(x\right)$

Similarly following the given problem,

$f ' \left(x\right) = \frac{1}{\tan} x \cdot {\sec}^{2} x$

$f ' \left(x\right) = \cos \frac{x}{\sin} x \cdot \frac{1}{{\cos}^{2} x}$

$f ' \left(x\right) = \frac{1}{\sin x \cos x}$

for simplifying further, we multiply and divide by 2,

$f ' \left(x\right) = \frac{2}{2 \sin x \cos x}$

$f ' \left(x\right) = \frac{2}{\sin 2 x}$

$f ' \left(x\right) = 2 \left(\cos e c 2 x\right)$