Question

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

(A) $(-\infty, 0) \cup(0, \infty)$

(B) $(-\infty,-1) \cup(-1, \infty)$

(C) $(-\infty, \infty)$

(D) $(0, \infty)$

Answer 1

$f(x)=\left\{\begin{array}{ll}x /(1-x), & x<0 \\ x /(1+x), & x \geq 0\end{array}\right.$

$\Rightarrow f^{\prime}(x)=\left\{\begin{array}{ll}1 /(1-x)^2, & x<0 \\ 1 /(1+x)^2, & x \geq 0\end{array}\right.$

Hence, $f(x)$ exist every where

So, The correct option will be (C).