Answer
$f'(x) = \dfrac{-9\cot ^8 \theta \csc ^2 \theta}{2\sqrt{\cot ^9 \theta +1}}$
Work Step by Step
In order to derivate this function you have to apply the chain rule
Let's make an «u» substitution to make it easier
$u = \cot ^9 \theta +1$
$f(u) = \sqrt{u} $
Derivate the function:
$f'(u) = \dfrac{u'}{2\sqrt{u}} $
Now let's find u'
$u' = -9\cot ^8 \theta \csc ^2 \theta$
Then undo the substitution, simplify and get the answer:
$f'(x) = \dfrac{-9\cot ^8 \theta \csc ^2 \theta}{2\sqrt{\cot ^9 \theta +1}}$
