How to show that the first derivative is bounded in a function

Question

\begin{equation} y=\arccos\left( -\frac{1}{2\left(Dr^{\dfrac {|\sin(2x+\theta)|}{M\sin x\sqrt{A+2B\cos(2x+\theta)}}}+1\right)} \right) \nonumber \end{equation}

How to show in above function the first derivative is bounded?

$\theta,D,M,A,B,r$ are constants but they have bounds too

$\theta<2\pi/3$,

$A,B,M,D>0$

$1\leq r \leq 2$

Also for the variables

$y \geq 0$ and $\pi/2 \leq x \leq 2\pi/3$