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

If $\frac{x^2}{\mathrm{a}^2}+\frac{y^2}{\mathrm{~b}^2}=1$, then $\frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}$ is