If the function f  is defined by  $f\left(x\right)=\frac{x}{1+\left|x\right|}$ then at what points is f differentiable 

If x>0,  $f\left(x\right)=\frac{x}{1+x}$ is differentiable
If x<0, $f\left(x\right)$ is differentiable
At x=0,  $f\text{'}\left(0^{-}\right)=\underset{x\to 0}{lt}\frac{\frac{x}{1-x}-0}{x-0}=\underset{x\to 0}{\lim }\frac{1}{1-x}=1$
$\begin{array}{l}f\text{'}\left(o^{+}\right)=\underset{x\to 0}{lt}\frac{\frac{x}{1+x}-0}{x-0}=\underset{x\to 0}{\lim }\frac{1}{1+x}=1\\ \therefore fisdiffatx=\text{0\hspace{0.17em} \&\hspace{0.17em}hence\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}everywhere}\end{array}$