Differentiate the following w.r.t. x:

Let $y=\log (\log x)$

By using the chain rule, we obtain

$\frac{d y}{d x}=\frac{d}{d x}[\log (\log x)]$

$=\frac{1}{\log x} \cdot \frac{d}{d x}(\log x)$

$=\frac{1}{\log x} \cdot \frac{1}{x}$

$=\frac{1}{x \log x}, x>1$