Question

Let $f(x)=2 \sin ^{-1} \sqrt{1-x}+\sin ^{-1}(2 \sqrt{x(1-x)})$ and $g(x)=\cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right)$ then derivative of $f(x)$ w.r.t. $g(x)$ at $x=\frac{1}{4}$ is

• A. $\frac{17}{64}$
• B. $\frac{17}{32}$
• C. 0
• D. None

Answer 1

Answer: (C)

Put $x=\cos ^2 \theta$
$f ( x )=2 \theta+\sin ^{-1}(\sin 2 \theta)=2 \theta+\pi-2 \theta=\pi $
$\therefore f ^{\prime}\left(\frac{1}{4}\right)=0$
Hence, derivative of $f(x)$ w.r.t. $g(x)$ at $x=\frac{1}{4}$ is 0 .