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

$\text{ put }x=1\Rightarrow y=\frac{\pi }{2}\text{ and then differentiate }$

$u=x^{2x}$

$\begin{array}{l}\frac{du}{dx}=x^{2x}(2\log x+2)\\ v=x^x\\ \frac{dv}{dx}=x^x(1+\log x)\\ x^{2x}-2x^x\cot y-1=0\end{array}$

 Differentiation w.r.to ‘x’

$\begin{array}{l}{x}^{2x}(2\log x+2)-2x^x(1+\log x)\cot y+2x^x{\mathrm{cosec}}^{2}y{y}^1=0\\ \Rightarrow 1\left(2\right(0)+2)-2(1+0)\left(0\right)+2y^1=0\\ \Rightarrow 2+2y^1=0\\ \Rightarrow y^1=\frac{-2}{2}\\ \Rightarrow y^1=-1\end{array}$