What is $\frac{dy}{dx}$ for $y = \cos^{-1}\left(2x^{2} - 1\right)$, $0 < x < 1$?

Explanation

1. 01

   Put $x = \cos \theta$: $2\cos^{2} \theta - 1 = \cos 2\theta$, so the function is $2\theta = 2\cos^{-1} x$.

2. 02

   So $\frac{dy}{dx} = - \frac{2}{\sqrt{1 - x^{2}}}$.