For $x\in R,f\left(x\right)=|{\log }_{10}2-\sin x|$ and  g(x) = f(f(x)), then:

In the neighbourhood of x = 0, f(x) = log 2 – sin x
$\therefore$ g(x) = f(f(x)) = log2 – sin (f(x))
log 2 – sin(log 2 – sin x)
Since g(x) is differentiable at x = 0,  $\therefore g\text{'}$ (x) = – cos(log2 – sinx) (–cosx)

$\Rightarrow g\text{'}\left(0\right)=\cos \left(\log 2\right)$