If $\mathrm{f}(x)=\sqrt{1+\cos ^2\left(x^2\right)}$, then $\mathrm{f}^{\prime}\left(\frac{\sqrt{\pi}}{2}\right)$ is

If $\mathrm{f}(x)=\frac{\sin ^2 x}{1+\cot x}+\frac{\cos ^2 x}{1+\tan x}$, then the value of $\mathrm{f}^{\prime}\left(\frac{\pi}{6}\right)$ is equal to

If $y=\tan ^{-1}\left(\frac{12 x-64 x^3}{1-48 x^2}\right)$, then $\frac{\mathrm{d} y}{\mathrm{~d} x}=$

If

$$ \sqrt{y-\sqrt{y-\sqrt{y-\ldots \ldots \ldots \infty}}}=\sqrt{x+\sqrt{x+\sqrt{x+\ldots \ldots \ldots \infty}}} $$

then $\frac{\mathrm{d} y}{\mathrm{~d} x}=$