Differentiability of $\cos |x-5|+\sin |x-3|+|x+10|^3-(|x|+4)$. Question : Test the differentiability of the function $$\cos |x-5|+\sin |x-3|+|x+10|^3-(|x|+4)$$ at the points $5, 3, -10, 0$.
Solution : Now $\cos |x|$ is differentiable everywhere. So is $\sin |x|$ as well as $|x|^3$. Therefore the only issue is the function $|x|$ at $x=0$. Therefor the function is differentiable at all the other points except $x=0$. 
Is this correct?

No, it is not correct. First, $\cos |x-5|$ does not vanish at $5$. Remember $\cos 0=1\neq 0$.   
$\cos |x|$ is differentiable everywhere.
$\sin |x|$ is differentiable everywhere except at $x=0$.
More generally, $|x|^n$ is differentiable everywhere for any $n\in\Bbb Z_{\ge 2}$.   
$|x|$ is differentiable everywhere but at $x=0$.   
So the function is differentiable everywhere but at the points $x$ such that either $x-3=0$ or $x=0$.
