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(C) Given $f(x) = |\log 2 - \sin x|$. Since $\log 2 \approx 0.693$ and $\sin x$ is small near $x=0$,$\log 2 - \sin x > 0$ for $x$ near $0$. Thus,$f(x)

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$g^{\prime}(0) = \cos(\log 2)$

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(C) Given $f(x) = |\log 2 - \sin x|$. Since $\log 2 \approx 0.693$ and $\sin x$ is small near $x=0$,$\log 2 - \sin x > 0$ for $x$ near $0$. Thus,$f(x) = \log 2 - \sin x$ for $x$ in a neighborhood of $0$. Then $g(x) = f(f(x)) = \log 2 - \sin(\log 2 - \sin x)$. Since $g(x)$ is a composition of differentiable functions near $x=0$,it is differentiable at $x=0$. Now,$g^{\prime}(x) = -\cos(\log 2 - \sin x) \cdot (-\cos x) = \cos(\log 2 - \sin x) \cdot \cos x$. Evaluating at $x=0$: $g^{\prime}(0) = \cos(\log 2 - \sin 0) \cdot \cos 0 = \cos(\log 2) \cdot 1 = \cos(\log 2)$.

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

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