Answer
$\left( \cos x\right) ^x\left( \ln \cos x-x\tan x\right) $
Work Step by Step
$\dfrac {d}{dx}\left( \left( \cos x\right) ^x\right) =x\times \left( \cos x\right) ^{x-1}\times \left( \dfrac {d}{dx}\left( \cos x\right) \right) +\left( \cos x\right) ^x\ln \cos x=\left( \cos x\right) ^x\left( \ln \cos x-x\tan x\right) $
