Question

The derivative of $\cos^{-1} (2x^2 - 1)$ w.r.t $\cos^{-1} x$ is

• A. $2$
• B. $\frac{-1}{2\sqrt{1 -x^2}}$
• C. $\frac{2}{x}$
• D. $1 - x^2$

Answer 1

Answer: (A)

$2$

Answer 2

Let $u=\cos ^{-1}\left(2 x^{2}-1\right)$ and $v=\cos ^{-1} \,x$
Now, $u=\cos ^{-1}\left(2 x^{2}-1\right)$
Put $x=\cos \theta$
$\therefore \, u=\cos ^{-1}\left(2 \cos ^{2} \theta-1\right)$
$=\cos ^{-1}(\cos 2 \theta)$
$=2 \theta=2 \cos ^{-1} \,x$
Again, $\frac{d u}{d v}=\frac{\left(\frac{d u}{d x}\right)}{\left(\frac{d v}{d x}\right)}$
$=\frac{\frac{d}{d x}\left(2 \cos ^{-1} x\right)}{\frac{d}{d x}\left(\cos ^{-1} x\right)}$
$=\frac{\left(-\frac{2}{\sqrt{1-x^{2}}}\right)}{\left(\frac{-1}{\sqrt{1-x^{2}}}\right)}=2$