f(x)=sin−1⁡2x1+x2 is differentiable on

f(x)=sin−12x1+x2⇒f'x=11−2x1+x22ddx2x1+x2 = 1+x21+x22−4x2.1+x2.2−2x2x1+x22 = 11−x2221−x21+x2 = 21+x2 if x 1∴f'x does not exist at x=1, i.e x=±1∴f(x) is differentiable on R−−1,1.

f(x)=sin−1⁡2x1+x2 is differentiable on

f(x)=sin−12x1+x2⇒f'x=11−2x1+x22ddx2x1+x2 = 1+x21+x22−4x2.1+x2.2−2x2x1+x22 = 11−x2221−x21+x2 = 21+x2 if x 1∴f'x does not exist at x=1, i.e x=±1∴f(x) is differentiable on R−−1,1.

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$f (x) = \sin^{- 1} (\frac{2 x}{1 + x^{2}}) is differentiable on$

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$R - [- 1, 1]$

$[- 1, 1]$

none of these

$R + [- 1, 1]$

answer is B.

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Detailed Solution

$\begin{array}{l} f (x) = \sin^{- 1} (\frac{2 x}{1 + x^{2}}) \\ \Rightarrow f' (x) = \frac{1}{\sqrt{1 - {(\frac{2 x}{1 + x^{2}})}^{2}}} \frac{d}{d x} (\frac{2 x}{1 + x^{2}}) \\ = \frac{1 + x^{2}}{\sqrt{{(1 + x^{2})}^{2} - 4 x^{2}}} . \frac{(1 + x^{2}) .2 - 2 x (2 x)}{{(1 + x^{2})}^{2}} \\ = \frac{1}{\sqrt{{(1 - x^{2})}^{2}}} \frac{2 (1 - x^{2})}{1 + x^{2}} \\ = \{\begin{cases} \frac{2}{1 + x^{2}} i f |x| < 1 \\ \frac{- 2}{1 + x^{2}} i f |x| > 1 \end{cases} \\ ∴ f' (x) d oes not exist at |x| = 1, i . e x = \pm 1 \\ ∴ f (x) is differentiable on R - \{- 1, 1\} . \end{array}$