If f(x) = loga loga(x), then f'(x) is :

$f'(x)=\frac{1}{\log _a x} .\left(\log _a e\right) . \frac{d}{d x}\left(\log _a x\right)$

$=\frac{\log _a e}{\log _a x} . \frac{1}{x} . \log _a e=\frac{\log _a e}{x} . \frac{\log _a e}{\log _a x}=\frac{\log _a e}{x \log _e x}$

Hence (1) is correct answer.