If y= ^ -1 (1 2 x^2-1 ) then Find d y d x . - Vidyadip

Question

If $y=\sec ^{-1}\left(\frac{1}{2 x^2-1}\right)$ then Find $\frac{d y}{d x}$.

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Answer

Using the property $\sec ^{-1}(1 / \theta)=\cos ^{-1}(\theta)$, we get: $y=\cos ^{-1}\left(2 x^2-1\right)$.
Substitute $x=\cos \theta \Longrightarrow \theta=\cos ^{-1} x$.
$y=\cos ^{-1}\left(2 \cos ^2 \theta-1\right)=\cos ^{-1}(\cos 2 \theta)=2 \theta$.
Substitute $\theta$ back: $y=2 \cos ^{-1} x$.
Differentiate with respect to $x: \frac{d y}{d x}=2 \cdot\left(\frac{-1}{\sqrt{1-x^2}}\right)$.
$\frac{d y}{d x}=\frac{-2}{\sqrt{1-x^2}}$.

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