Mathematics - Continuity - Differentiability Question with Solution | TestHub

In the neighborhood of$x=0, f\left(x\right)=\log 2-\sin x$

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

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

$\therefore$$g^{\prime }\left(x\right)=-\cos \left(\log 2-\sin x\right) \left(–\cos x\right)$

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