Differentiate the following w.r.t x: $\tan^{-1}\bigg(\large {\frac {a \cos x - b \sin x}{b \cos x + a \sin x}}\bigg)$, where $\frac{-\pi}{2} < x < \frac{\pi}{2}$ and $\frac{a}{b} \tan x > -1$

Question

$\begin{array}{1 1} 1 \\ \large\frac{1}{2} \\ -1 \\ \large\frac{-1}{2} \end{array}$