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Q. If $f(x) = \sin^{-1} \left(\frac{2 \times3^{x}}{1 +9^{x}}\right) $, then $f' \left(- \frac{1}{2}\right) $ equals :

JEE MainJEE Main 2018Continuity and Differentiability

Solution:

Given:$f(x)=\sin ^{-1}\left(\frac{2.3^{x}}{1+\left(3^{x}\right)^{2}}\right)$
Let $3^{x}=\tan \theta$, then
$f(\theta)=\sin ^{-1}\left(\frac{2\, \tan \theta}{1+\tan ^{2} \theta}\right)$
$\Rightarrow f(\theta)=\sin ^{-1}(\sin 2 \theta)=2 \theta$
$ \Rightarrow f(x)=2 \tan ^{-1}\left(3^{2}\right)$
So $f'(x)=2 \frac{1}{\left(1+9^{x}\right)} \cdot 3^{x} \ln 3$
$f'\left(\frac{-1}{2}\right)=\frac{2}{1+9^{-1 / 2}} 3^{-1 / 2} \ln 3=\sqrt{3} \ln \sqrt{3}=\sqrt{3} \log _{e} \sqrt{3}$