Question

How to prove f(x)=x/(1+|x|) is differentiable at every point

Answer 1

Dear student,
The only point at which the function can be non differentiable is x=0
So RHD at x=0
$$\begin{gathered} RHD=\underset{h\to 0}{lim}\frac{f(0+h)-f\left(0\right)}{h} \\ =\underset{h\to 0}{lim}\frac{{\displaystyle \frac{h}{1+h}}-0}{h} \\ =\underset{h\to 0}{lim}\frac{1}{1+h} \\ =1 \end{gathered}$$
LHD at x=0

$$\begin{gathered} LHD=\underset{h\to 0}{lim}\frac{f(0-h)-f\left(0\right)}{-h} \\ =\underset{h\to 0}{lim}\frac{{\displaystyle \frac{-h}{1+h}}}{-h} \\ =\underset{h\to 0}{lim}\frac{1}{1+h} \\ =1 \end{gathered}$$
Since LHD =RHD
The function is derivable at x=0 and hence is derivable for all x
Regards