Answer
$\dfrac{1}{2 x \ln x \sqrt {\ln (\ln x)}}$
Work Step by Step
We have: $S(x)= \sqrt {\ln (\ln x)}$
We differentiate both sides with respect to $x$.
$S^{\prime}(x)=\dfrac{d}{dx} [ \sqrt {\ln (\ln x)}] \\=\dfrac{1}{2 \sqrt {\ln (\ln x)}} \times \dfrac{d}{dx} [\ln (\ln x)]\\=\dfrac{1}{2 \sqrt {\ln (\ln x)}} \times \dfrac{1}{\ln (x)} \times \dfrac{1}{x}$
Simplify to obtain:
$S^{\prime}(x)=\dfrac{1}{2 x \ln x \sqrt {\ln (\ln x)}}$
