The set of points where $f(x)=\frac{x}{1+|x|}$ is differentiable is:

$f(x)=\frac{x}{1-x}$ for $x<0$ and $f(x)=\frac{x}{1+x}$ for $x\ge 0$.For $x<0$: $f^{\prime }(x)=\frac{(1-x)(1)-x(-1)}{(1-x{)}^2}=\frac{1}{(1-x{)}^2}$.As $x\to 0^{-}$, $f^{\prime }(x)\to 1$.For $x>0$: $f^{\prime }(x)=\frac{(1+x)(1)-x(1)}{(1+x{)}^2}=\frac{1}{(1+x{)}^2}$.As $x\to 0^{+}$, $f^{\prime }(x)\to 1$.Since LHD $=$ RHD at $x=0$, the function is differentiable at $x=0$.The denominator $1+|x|$ is never zero for real $x$.Thus, $f(x)$ is differentiable everywhere on $(-\infty ,\infty )$.