The domain of the function defined by $$f(x)=\cos ^{-1} \sqrt{x-1}$$ is

The value of $$\cos \left(\sin ^{-1} \frac{\pi}{3}+\cos ^{-1} \frac{\pi}{3}\right)$$ is Does not exist

If $$f(x)=\sin ^{-1}\left(\frac{2^{x+1}}{1+4^x}\right)$$ then $$f^{\prime}(0)=$$

$$\cos \left[2 \sin ^{-1} \frac{3}{4}+\cos ^{-1} \frac{3}{4}\right]=$$