# Question #7f6d8

Oct 13, 2017

I personally like to rewrite functions to make it easier to use the product rule instead of the quotient rule where applicable.

Let $f \left(x\right) = {\cos}^{2} \frac{x}{\sqrt{\cos 2 x}}$

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

Here we can see we will use the chain rule and product rule to differentiate.

$\frac{d}{\mathrm{dx}} f \left(x\right) = - 2 \cos \left(x\right) \sin \left(x\right) \cdot {\left(\cos \left(2 x\right)\right)}^{- \frac{1}{2}} + {\cos}^{2} \left(x\right) \left(- \frac{1}{2}\right) \cos {\left(2 x\right)}^{- \frac{3}{2}} \left(- \sin \left(2 x\right)\right) \left(2\right)$

Simplify:

$= \frac{{\cos}^{2} \left(x\right) \sin \left(2 x\right)}{\cos {\left(2 x\right)}^{\frac{3}{2}}} - \frac{2 \cos \left(x\right) \sin \left(x\right)}{\sqrt{\cos} \left(2 x\right)}$

(I moved the negative fraction to the back to avoid having a negative up front)

You could simplify this further if you wanted just one fraction, you can just use your arithmetic skills for that.