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Q.
If the function $f$ is defined by $f(x)=\frac{x}{1+|x|}$ then at what points is $f$ differentiable
Solution:
if $x>0$, then $f(x)=\frac{x}{1+x}$ is differentiable
$f'\left(0^{-}\right)=\displaystyle\lim _{x \rightarrow 0} \frac{\frac{x}{1+x}-0}{x-0}$
$=\displaystyle\lim _{x \rightarrow 0} \frac{1}{1-x}=1$
if $x<0$, then $f(x)=\frac{x}{1-x}$ is differentiable at $x=0$,
$f'\left(0^{+}\right)=\displaystyle\lim _{x \rightarrow 0} \frac{\frac{x}{1+x}-0}{x-0}$
$=\displaystyle\lim _{x \rightarrow 0} \frac{1}{1+x}=1$
$f(x)$ is differentiable at $x=0$ and hence every where