Differentiate the following functions with respect to $x$ :
$\sin ^{-1}\left(1-2 x^{2}\right), 0
$y=\sin ^{-1}\left\{1-2 x^{2}\right\}$
let $x=\sin \theta$
Now
$y=\sin ^{-1}\left\{\sqrt{1-2 \sin ^{2} \theta}\right\}$
Using $1-2 \sin ^{2} \theta=\cos 2 \theta$
$y=\sin ^{-1}(\cos 2 \theta)$
$y=\sin ^{-1}\left\{\sin \left(\frac{\pi}{2}-2 \theta\right)\right\}$
Considering the limits,
$0 $0<\sin \theta<1$ $0<\theta<\frac{\pi}{2}$ $0<2 \theta<\pi$ $0>-2 \theta>-\pi$ $\frac{\pi}{2}>\frac{\pi}{2}-2 \theta>-\frac{\pi}{2}$ Now, $y=\sin ^{-1}\left\{\sin \left(\frac{\pi}{2}-2 \theta\right)\right\}$ $y=\frac{\pi}{2}-2 \theta$ $y=\frac{\pi}{2}-2 \sin ^{-1} x$ Differentiating w.r.t $x$, we get $\frac{d y}{d x}=\frac{d}{d x}\left(\frac{\pi}{2}-2 \cos ^{-1} x\right)$ $\frac{\mathrm{dy}}{\mathrm{dx}}=0-2\left(\frac{1}{\sqrt{1-\mathrm{x}^{2}}}\right)$ $\frac{d y}{d x}=\frac{-2}{\sqrt{1-x^{2}}}$