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

$|x|$ is non-differentiable function at $$\begin{aligned}& x=0 \text{ as L.H.D }=-1 \text{ and R.H. } \mathrm{D}=1 \\ & \because|x|=\left\{\begin{array}{cl} x, & x \geq 0 \\ -x, & x < 0 \end{array}\right. \end{aligned}$$) But (\cos |h|) is differentiable (\therefore) Any combination of two such functions will be non-differentiable. Hence option (a) and (b) are ruled out. Now, consider (\sin |x|+|x|) ($$\begin{aligned} & L^{\prime}=\lim _{h \rightarrow 0} \frac{\sin |-h|+|-h|}{-h}=\lim _{h \rightarrow 0} \frac{\sin h}{-h}-1=-1-1 =-2 \\ \text{ and } R^{\prime}=\lim _{h \rightarrow 0} \frac{\sin |h|+|h|}{h}=\lim _{h \rightarrow 0} \frac{\sin h}{h}+1=1+1=2 \\ \text{ Consider } \sin |x|-|x| \\ & L^{\prime}=\lim _{h \rightarrow 0} \frac{\sin |-h|-|-h|}{-h}=\lim _{h \rightarrow 0} \frac{\sin h}{-h}+1=0 \\ & \end{aligned}$$)