Which of the following functions is differentiable at x = 0.

$f\left(x\right)=\cos \left(\right|x\left|\right)+\left|x\right|=\cos x+\left|x\right|$

We know that $\left|x\right|$is non–differentiable at x = 0.

So,f(x) is  non–differentiable at $x = 0$

Similarly $f\left(x\right)=\cos \left(\right|x\left|\right)-\left|x\right|=\cos x-\left|x\right|\text{ is }$also

non–differentiable at x = 0

$$\begin{gathered} f\left(x\right)=\sin \left|x\right|+\left|x\right|=\left\{\begin{array}{l}-\sin x-x,x<0\\ +\sin x+x,x\ge 0\end{array}\right. \\ Here f\text{'}\left(0^{-}\right)\ne f\text{'}\left(0^{+}\right) \end{gathered}$$

$\therefore f\left(x\right)\text{ is not differentiable at }x=0\text{. }$

$\Rightarrow f\left(x\right)=\sin \left|x\right|-\left|x\right|=\left\{\begin{array}{l}-\sin x+x,x<0\\ \sin x-x,x\ge 0\end{array}\right.$
$$\begin{gathered} \therefore f^{\mathrm{\prime }}\left(x\right)=\left\{\begin{array}{l}-\cos x+1,x<0\\ +\cos x-1,x>0\end{array}\right. \\ Here f\text{'}\left(0^{-}\right)=f\text{'}\left(0^{+}\right)=0 \\ So,f is differentiable at x=0 \end{gathered}$$