$\mathrm{Let} \text{'}\mathrm{y}\text{'} \mathrm{be} \mathrm{an} \mathrm{implicit} \mathrm{function} \mathrm{of} \mathrm{x} \mathrm{defined} \mathrm{by} {\mathrm{x}}^{2\mathrm{x}}-2{\mathrm{x}}^{\mathrm{x}} \cot \mathrm{y}-1=0. \mathrm{Then} {\mathrm{y}}^1\left(1\right) \mathrm{equals}$

$$\begin{gathered} {\mathrm{x}}^{2\mathrm{x}}-2·{\mathrm{x}}^{\mathrm{x}} \cot \mathrm{y}-1=0 \to \left(1\right) \\ {\left({\mathrm{x}}^{\mathrm{x}}-\cot \mathrm{y}\right)}^2=1+{\cot }^2\mathrm{y} \\ ={\mathrm{cosec}}^2\mathrm{y} \\ {\mathrm{x}}^{\mathrm{x}}-\cot \mathrm{y}=\mathrm{cosec} \mathrm{y} \\ {\mathrm{x}}^{\mathrm{x}}=\mathrm{cosec} \mathrm{y}+\cot \mathrm{y} \\ =\frac{1+\cos \mathrm{y}}{\sin \mathrm{y}} \\ {\mathrm{x}}^{\mathrm{x}}=\cot \raisebox{1ex}{$\mathrm{y}$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \\ {\mathrm{x}}^{\mathrm{x}}(1+\log \mathrm{x})=-{\mathrm{cosec}}^2\frac{\mathrm{y}}{2}·\frac{1}{2} \frac{\mathrm{dy}}{\mathrm{dx}} \to \left(2\right) \end{gathered}$$

$$\begin{gathered} \mathrm{From} \left(1\right) \\ \mathrm{x}=1 \Rightarrow 1-2 \cot \mathrm{y}-1=0 \\ \cot \mathrm{y}=0 \\ \mathrm{y}=\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \\ \left(2\right) \\ \Rightarrow 1(1+0)=-{\left(\sqrt{2}\right)}^2\frac{1}{2}·\frac{\mathrm{dy}}{\mathrm{dx}} \\ \frac{\mathrm{dy}}{\mathrm{dx}}=-1 \end{gathered}$$