Step 1:

$f(x)=\mid x+2\mid$

$f(x)=\left\{\begin{array}{1 1}-(x+2),&for\;x< -2\\(x+2)&for\;x\geq-2\end{array}\right.$

Let us check the differentiability at $x=-2$

LHD at $x=1=\lim\limits_{x\to 2^-}\large\frac{f(x)-f(-2)}{x+2}$

$\Rightarrow \lim\limits_{x\to 2^-}\large\frac{-(x+2)-0}{x+2}$

$\Rightarrow \lim\limits_{x\to 2^-}\large\frac{-x-2}{x+2}=\large\frac{-(x+2)}{x+2}$

$\Rightarrow -1$

Step 2:

RHD at $x=-2$ is

$\lim\limits_{x\to 2^+}\large\frac{f(x)-f(-2)}{x+2}$

$\Rightarrow 1$

Hence LHD $\neq$ RHD

Hence f(x) is not differentiable at $x=-2$