Solving Trigonometric Derivatives I have a function $F(θ) = \sin^{−1} \sqrt{\sin(11θ)}$
I derived the following answer using basic trigonometric and quotient rules.
$\dfrac{11\csc \left(\left(11\theta \right)^{\left(\frac{1}{2}\right)}\right)\cos \left(\left(11\theta \right)\right)}{2\sqrt{1-\sin \left(11\theta \right)}}$
My answer however is wrong.
Can anyone outline how to go about getting the correct solution to a problem such as this?

We have $\sin F(x)= (\sin 11x)^{1/2}.$ so by the Chain Rule, $$(1)....(\cos F(x)\;)\cdot F'(x)=(1/2)(\sin 11x)^{(1/2-1)})(11\cos 11 x)=\frac {11\cos 11x}{2\sqrt {\sin 11x}}.$$  Assuming that we are using the  main (or usual) branch of $\sin^{-1}$, which takes values in $[-\pi /2,\pi /2],$ we have $F(x)\in [-\pi /2,\pi /2]$ so $\cos F(x)\geq 0$. 
So $$\cos F(x)=\sqrt {1-\sin^2F(x)}=\sqrt {1-\sin 11x}$$ which we can substitute  into (1).
