$\vert f(x)\vert$ is differentiable at $x=a$ then $f(x)$ is also differentiable at $x=a$ 
$\vert f(x)\vert$ is differentiable at $x=a$ then $f(x)$ is also differentiable at $x=a$

Is the above statement always true? If not, can you give an example?
$|x|$ and $x$ are both differentiable at $x=2$. The reverse is not always true. The example is $|x|$ itself. $y=x$ is differntiable at $x=0$ but $|x|$ is not.

Hint: Consider the function $$f(x)=\begin{cases}1 & x<=0\\ -1 & x>0.\end{cases}$$
