Is the function $\sin(\lvert x\rvert)$ differentiable? It is readily shown that the function $\sin(\lvert x\rvert)$ is differentiable when $x\ne 0$. What I know is that $\lvert x\rvert$ is not differentable at $x=0$....--prophetes.ai

Is the function $\sin(\lvert x\rvert)$ differentiable? It is readily shown that the function $\sin(\lvert x\rvert)$ is differentiable when $x\ne 0$. What I know is that $\lvert x\rvert$ is not differentable at $x=0$. But does $\sin(|x|)$ also follow the same rule? I believe that it is also not differentiable but why?

It is not at 0.

$x\to 0^+$ 1, $x\to 0^-$ -1