$\frac{d}{dx}(\sin^{-1}(3x - 4x^3)) = $

Let $S$ be the set of all solutions of the equation $\cos ^{-1}(2 x)-2 \cos ^{-1}\left(\sqrt{1-x^2}\right)=\pi$,where $x \in\left[-\frac{1}{2}, \frac{1}{2}\right]$. Then $\sum_{x \in S} 2 \sin ^{-1}\left(x^2-1\right)$ is equal to

If $\cos ^{-1} x-\cos ^{-1} \frac{y}{3}=\alpha$,where $-1 \leq x \leq 1$,$-3 \leq y \leq 3$,and $x \leq \frac{y}{3}$,then for all $x, y$,$9 x^2-6 x y \cos \alpha+y^2$ is equal to

The derivative of $\cos ^{-1}\left(2 x^{2}-1\right)$ with respect to $\cos ^{-1} x$ is

If $\cot ^{-1}(\alpha)=\cot ^{-1} 2+\cot ^{-1} 8+\cot ^{-1} 18+\cot ^{-1} 32+\ldots$ up to $100$ terms,then $\alpha$ is

If $\frac{(x+1)^{2}}{x^{3}+x}=\frac{A}{x}+\frac{Bx+C}{x^{2}+1}$,then $\sin^{-1} A + \tan^{-1} B + \sec^{-1} C$ is equal to