If f(x)=sin ^{-1}left(frac{2 x}{1+x^{2}}right | vedclass

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(B) We have,$f(x)=\sin ^{-1}\left(\frac{2 x}{1+x^{2}}ight)$.Let $x=	an 	heta$,then $	heta=	an ^{-1} x$.Substituting this into the function,we ge

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$\frac{1}{2}$

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(B) We have,$f(x)=\sin ^{-1}\left(\frac{2 x}{1+x^{2}}ight)$.Let $x=	an 	heta$,then $	heta=	an ^{-1} x$.Substituting this into the function,we get $f(x)=\sin ^{-1}\left(\frac{2 	an 	heta}{1+	an ^{2} 	heta}ight)$.Since $\sin 2	heta = \frac{2 	an 	heta}{1+	an ^{2} 	heta}$,we have $f(x)=\sin ^{-1}(\sin 2 	heta) = 2 	heta = 2 	an ^{-1} x$.Differentiating with respect to $x$,we get $f^{\prime}(x) = \frac{2}{1+x^{2}}$.Substituting $x=\sqrt{3}$,we get $f^{\prime}(\sqrt{3}) = \frac{2}{1+(\sqrt{3})^{2}} = \frac{2}{1+3} = \frac{2}{4} = \frac{1}{2}$.

If $f(x)=\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)$,then $f^{\prime}(\sqrt{3})$ is

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