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  4. If f(x)=(s i n)- 1((2 × 3x/1 + 9x)), then f'(- (1/2)) is equal to

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Q. If $f\left(x\right)=\left(s i n\right)^{- 1}\left(\frac{2 \times 3^{x}}{1 + 9^{x}}\right),$ then $f^{'}\left(- \frac{1}{2}\right)$ is equal to

NTA AbhyasNTA Abhyas 2020

Solution:

Given $f(x)=\sin ^{-1}\left(\frac{2 \times 3^{x}}{1+9^{x}}\right)$
Let $3^{x}=\tan (t) \Rightarrow t=\tan ^{-1}\left(3^{x}\right)$
So, $f(x)=\sin ^{-1}\left(\frac{2 \tan (t)}{1+\tan ^{2}(t)}\right)$
We know that, $\sin (2 t)=\frac{2 \tan (t)}{1+\tan ^{2}(t)}$
$\Rightarrow f(x)=\sin ^{-1}(\sin (2 t))$
$\therefore f(x)=2 t=2 \tan ^{-1}\left(3^{x}\right)$
$\Rightarrow \frac{d f(x)}{d x}=\frac{2}{1+\left(3^{x}\right)^{2}} \times 3^{x} \cdot \log _{e} 3$
At $x=\frac{1}{2}, \frac{d f}{d x}=\frac{2}{1+\left(3^{\frac{1}{2}}\right)^{2}} \times 3^{\frac{1}{2}} \cdot \log _{e} 3$
$=\frac{1}{2} \times \sqrt{3} \times \log _{e} 3$
$=\sqrt{3} \times \log _{e} \sqrt{3}$