For $$x>1$$, if $$(2 x)^{2 y}=4 \mathrm{e}^{2 x-2 y}$$, then $$(1+\log 2 x)^2 \frac{\mathrm{d} y}{\mathrm{~d} x}$$ is equal to

If $$\mathrm{f}(x)=\mathrm{e}^x, \mathrm{~g}(x)=\sin ^{-1} x$$ and $$\mathrm{h}(x)=\mathrm{f}(\mathrm{g}(x))$$, then $$\frac{\mathrm{h}^{\prime}(x)}{\mathrm{h}(x)}$$ is

If $$y$$ is a function of $$x$$ and $$\log (x+y)=2 x y$$, then the value of $$y^{\prime}(0)$$ is

If $$x^{\mathrm{k}}+y^{\mathrm{k}}=\mathrm{a}^{\mathrm{k}}(\mathrm{a}, \mathrm{k}>0)$$ and $$\frac{\mathrm{d} y}{\mathrm{~d} x}+\left(\frac{y}{x}\right)^{\frac{1}{3}}=0$$, then $$\mathrm{k}$$ has the value