If $\mathrm{a}\left(4+x^2\right)=x$ and $y-x^3=\mathrm{a}^2$ then $\frac{\mathrm{d} y}{\mathrm{~d} x}$ at $x=1$ is $\qquad$

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

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