Thank you for reporting, we will resolve it shortly
Q.
Let $f:(0, \infty) \rightarrow(0, \pi)$ be a function defined by $f(x)=\cot ^{-1}(\ln x)$ and $g$ be the inverse function of $f$, then
Continuity and Differentiability
Solution:
$f(1)=\cot ^{-1}(0)=\frac{\pi}{2} \Rightarrow g \left(\frac{\pi}{2}\right)=1, f ^{\prime}( x )=\frac{-1}{ x \left(1+\ln ^2 x \right)} \Rightarrow f ^{\prime}(1)=-1$
and $f ^{\prime \prime}( x )=\frac{1}{ x ^2\left(1+\ln ^2 x \right)}+\frac{2 \ln x }{ x ^2\left(1+\ln ^2 x \right)} \Rightarrow f ^{\prime \prime}(1)=1$
$g ^{\prime}\left(\frac{\pi}{2}\right)=\frac{1}{ f ^{\prime}(1)}=-1 $
$g ^{\prime \prime}\left(\frac{\pi}{2}\right)=\frac{-\left( f ^{\prime \prime}(1)\right)}{\left( f ^{\prime}(1)\right)^3}=\frac{-1}{-1}=1$