For $x\in R,\ f(x)=|\log 2-\sin x|$ and $g(x)=f(f(x))$, then ;

In the neighourwood of $x=0$ $f(x)=\log 2-\sin x$

$g(x)=f(x)=\log 2-\sin f(x)$

$=\log 2-\sin (\log 2-\sin x)$

$g'(x)=\cos (\log2-\sin x)(-\cos x)$

$g'(0)=\cos (log 2)$