Find the derivative with respect to x of the function : log cos x  sin x log sin x  cos x -1 + arc…

$y=\left({\text{log}}_{\text{cos}x}\text{ sin}x\right){\left({\text{log}}_{\text{sin}x}\text{ cos}x\right)}^{-1}+arc\text{sin}\frac{2x}{1+x^2}$

$=\frac{\ell nsinx}{\ell ncosx}{\left(\frac{\ell ncosx}{\ell nsinx}\right)}^{-1}+2{tan}^{-1}x$

$y={\left(\frac{\ell nsinx}{\ell ncosx}\right)}^2+2{tan}^{-1}x$

$\Rightarrow$ $\frac{\text{dy}}{\text{d}x}=2\left(\frac{\ell nsinx}{\ell ncosx}\right)\frac{\left(\ell ncosx\right)\frac{cosx}{sinx}+\left(\ell nsinx\right)\frac{sinx}{cosx}}{{\left(\ell ncosx\right)}^2}+\frac{2}{1+x^2}$    [Using chain rule]

$\Rightarrow$$\frac{\text{dy}}{\text{d}x}{|}_{\frac{\pi }{4}}=2\frac{\left(1+1\right)}{{\left(\ell n\frac{1}{\sqrt{2}}\right)}^{}}+\frac{2}{1+{\left(\frac{\pi }{4}\right)}^2}$

$=\frac{2\times 2}{\left(\frac{-1}{2}\right)\ell n2}+\frac{2}{1+{\left(\frac{\pi }{4}\right)}^2}$

$=-\frac{8}{\ell n2}+\frac{32}{16+{\pi }^2}$