If $y = \tan^{-1} \left[ \frac{5 \cos x - 12 \sin x}{12 \cos x + 5 \sin x} \right]$,then $\frac{dy}{dx}$ is equal to

Find the value of $\frac{d}{dx} \left[ \tan^{-1} \sqrt{\frac{1 - \cos x}{1 + \cos x}} \right]$.

If $y = \operatorname{Tan}^{-1} \sqrt{x^2-1} + \operatorname{Sinh}^{-1} \sqrt{x^2-1}$,$x > 1$,then $\frac{dy}{dx} = $

Let $0 < x < \pi$ and $y(x)$ be given by $(1+\sin x)y^3 - (\cos x)y^2 + 2(1+\sin x)y - 2\cos x = 0$. The derivative of $y$ with respect to $\tan \frac{x}{2}$ at $x = \frac{\pi}{2}$ is:

The derivative of $\sec^{-1}\left( \frac{1}{2x^2 - 1} \right)$ with respect to $\sqrt{1 - x^2}$ at $x = \frac{1}{2}$ is:

If $y=\tan ^{-1}\left(\frac{4 \sin 2 x}{\cos 2 x-6 \sin ^2 x}\right)$,then $\left(\frac{d y}{d x}\right)$ at $x=0$ is