If $y=\tan ^{-1}\left\{\frac{a \cos x-b \sin x}{b \cos x+a \sin x}\right\}$,then find $\frac{d y}{d x}$.

If $\cot[f(x)] = \frac{3x - x^3}{1 - 3x^2}$ and $\sin[g(x)] = \frac{1 - x^2}{1 + x^2}$, then $\lim_{x \to t} \frac{f(x) - f(t)}{g(x) - g(t)} = \dots$

The differential coefficient of $\tan^{-1}\left( \frac{\sqrt{1+x} - \sqrt{1-x}}{\sqrt{1+x} + \sqrt{1-x}} \right)$ is

If $y=\cos ^{-1}\left(\frac{6 x-2 x^2-4}{2 x^2-6 x+5}\right)$, then $\frac{d y}{d x}=$

Differentiate the function with respect to $x$: $\cot ^{-1}\left[\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right]$,where $0 < x < \frac{\pi}{2}$.

If $y = \operatorname{Tan}^{-1}\left(\frac{2x}{1-x^2}\right)$ where $|x| < 1$,then find the value of $\left(\frac{dy}{dx}\right)$ at $x = \frac{1}{2}$.