Question

The derivative of $\cos ^{-1}\left(2 x^{2}-1\right)$ with respect to $\cos ^{-1} x$ is

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

Answer 1

Answer: (A)

Let $u=\cos ^{-1}\left(2 x^{2}-1\right)$
$\frac{d u}{d x}=\frac{-1}{\sqrt{1-\left(2 x^{2}-1\right)^{2}}} 4 x=\frac{-4 x}{\sqrt{4 x^{2}\left(1-x^{2}\right)}}$
$=\frac{-4 x}{2 x \sqrt{1-x^{2}}}=\frac{-2}{\sqrt{1-x^{2}}}$
$v=\cos ^{-1} x$
$\frac{d v}{d x}=-1 / \sqrt{1-x^{2}}$
$\frac{d u}{d v}=\frac{-2 / \sqrt{1-x^{2}}}{-1 / \sqrt{1-x^{2}}}=2$