If $u=\frac{\tan ^{-1} x}{\tan ^{-1} x+1}$ and $v=\tan ^{-1}\left(\tan ^{-1} x\right)$ then $\frac{d u}{d v}=$

If $\mathrm{y}=x^x+x^{\frac{1}{x}}$, then $\frac{\mathrm{d} y}{\mathrm{~d} x}$ is equal to

If $f(1)=1, f^{\prime}(1)=3$, then the derivative of $\mathrm{f}(\mathrm{f}(\mathrm{f}(x)))+(\mathrm{f}(x))^2$ at $x=1$ is

If $y=\log _{\mathrm{e}} x^3+3 \sin ^{-1} x+\mathrm{kx}^2$ and $y^{\prime}\left(\frac{1}{2}\right)=2 \sqrt{3}$, then $k=$