Question
Diffrentiate the following w. r. t. x.
$\cos ^{-1}\left(1-x^2\right)$
$\cos ^{-1}\left(1-x^2\right)$
Differentiating w.r.t. x, we get
$\begin{aligned} \frac{d y}{d x} & =\frac{d}{d x}\left[\cos ^{-1}\left(1-x^2\right)\right] \\ & =\frac{-1}{\sqrt{1-\left(1-x^2\right)^2}} \cdot \frac{d}{d x}\left(1-x^2\right) \\ & =\frac{-1}{\sqrt{1-\left(1-2 x^2+x^4\right)}} \cdot(0-2 x) \\ & =\frac{2 x}{\sqrt{2 x^2-x^4}} \\ & =\frac{2 x}{x \sqrt{2-x^2}}=\frac{2}{\sqrt{2-x^2}} .\end{aligned}$
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