Answer
$f'(\theta)=\tan(\theta)+\sec(\theta)(\theta(5\tan(\theta)+\sec(\theta))+5).$
Work Step by Step
$f(\theta)=g(\theta)+h(\theta)\rightarrow g(\theta)=5\theta(\sec(\theta))$; $h(\theta)=\theta(\tan(\theta))$
Product Rule $g'(\theta)=((u(\theta)(v(\theta))’=u’(\theta)v(\theta)+u(\theta)v’(\theta))$.
$u(\theta)=5\theta ;u’(\theta)=5 $
$v(\theta)=\sec(\theta) ;v’(\theta)=\sec(\theta)\tan(\theta) $
$g'(\theta)=5(\theta\sec(\theta)\tan(\theta)+\sec(\theta))$.
Product Rule $h'(\theta)=((u(\theta)(v(\theta))’=u’(\theta)v(\theta)+u(\theta)v’(\theta))$.
$u(\theta)=\theta ;u’(\theta)=1 $
$v(\theta)=\tan(\theta) ;v’(\theta)=\sec^2(\theta) $
$h'(\theta)=(\tan(\theta)+\theta(\sec^2(\theta)))$.
$f'(\theta)=g'(\theta)+h'(\theta)$
$=\tan(\theta)+\sec(\theta)(\theta(5\tan(\theta)+\sec(\theta))+5).$
