# What is the derivative of f(x)=log_2(cos(x)) ?

Mar 18, 2018

$- \tan \frac{x}{\ln} \left(2\right)$

#### Explanation:

$f \left(x\right) = {\log}_{2} \left(\cos \left(x\right)\right) = \ln \frac{\cos \left(x\right)}{\ln} \left(2\right)$

$\frac{1}{\ln} \left(2\right)$ is just a constant and can be ignored.

$\left(\ln \left(u\right)\right) ' = \frac{u '}{u}$

$u = \cos \left(x\right) , u ' = - \sin \left(x\right)$

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