If $y=2 x^{3 x}$, then $d y / d x$ at $x=1$ is

$\frac{d}{d x}\left[\cos ^2\left(\cot ^{-1} \sqrt{\frac{2+x}{2-x}}\right)\right]$ is

If $$y=a \sin x+b \cos x$$, then $$y^2+\left(\frac{d y}{d x}\right)^2$$ is a

If $$f(x)=1+n x+\frac{n(n-1)}{2} x^2+\frac{n(n-1)(n-2)}{6} x^3+\ldots+x^n$$, then $$f^n(1)$$ is equal to :