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