So the function is $cos(|x-5|)+sin(|x-3|)+(|x+10|)^{3}-(|x|+4)^{2}$. I could not check it by the graph method and limit method is so long. Is there any other way to check the differentiability. The points on which differentiability is to be checked are; $x=5$, $x=3$, $x=-10$ and $x=0$.
I have edited the question because I was thinking that it could be done with the help of graph but I couldn't. Please help.
$$f(x)=\cos(|x-5|)+\sin(|x-3|)+(|x+10|)^{3}-(|x|+4)^{2}$$ As $\cos(-x)=\cos (x)$, the function can be simplified to the following $$f(x)=\cos(x-5)+\sin(|x-3|)+(|x+10|)^{3}-(|x|+4)^{2}$$ Now, you have to look for critical points in the function ie value of $x$ which cause a component to flip sign. Clearly, the critical points are $x=3,-10,0$. Hence you will have to graph the function in $4$ separate intervals namely $(-\infty,-10],[-10,0],[0,3]$ and $(3,\infty)$. For each interval you can open the mod, add appropriate sign and graph the function.