Differentiate $\tan ^{-1} x$ with respect to $\cot ^{-1} x$ for $x \in R$.

Let $f : R \rightarrow R$ be a differentiable function such that $f(2) = 2$. Then the value of $\lim_{x \to 2} \int_{2}^{f(x)} \frac{4t^3}{x - 2} dt$ is

If $f(x)=\cot ^{-1}\left(\frac{x^x-x^{-x}}{2}\right)$,then $f^{\prime}(1)=$

$\frac{d}{dx} \left[ \tan^{-1} \left( \frac{\sqrt{x}(3 - x)}{1 - 3x} \right) \right] =$

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 $0 < x < \frac{\pi}{2}$.

Differentiation of $\tan ^{-1}\left(\frac{x}{\sqrt{1-x^2}}\right)$ with respect to $\sin ^{-1}\left(3 x-4 x^3\right)$ is $....$