Question

If $f\left(x\right)={\sin }^{-1}\left(\frac{2x}{1+x^2}\right),$ then $f\left(x\right)$ is differentiable on :

• A. $[-1,1]$
• B. $R-\{-1,1\}$
• C. $R-(-1,1)$
• D. none of these

Answer 1

The correct option is B $R-\{-1,1\}$

$f\left(x\right)={\sin }^{-1}\left(\frac{2x}{1+x^2}\right)$

is differentiable in the region $xϵR-\{-1,1\}$

We can draw its graph :

and we know that functions having sharp edges and gaps are not differentiable at that point. So here we have sharp edges at $x=-1$ and $x=1$. So not differentiable at $-1,1$. Rest at all points the function is differentiable.