If `y=sec^(-1)(1/(2x^2-1));0

Explanation

To find the derivative of the function \( y = \sec^{-1}\left(\frac{1}{2x^2 - 1}\right) \) for \( 0 < x < \frac{1}{\sqrt{2}} \), we can follow these steps:

### Step 1: Substitute \( x \) with \( \cos \theta \)
Let \( x = \cos \theta \). Then, we can express \( \theta \) as \( \theta = \cos^{-1}(x) \).

### Step 2: Rewrite \( y \)
Substituting \( x \) into the equation for \( y \):
\[ y = \sec^{-1}\left(\frac{1}{2(\cos \theta)^2 - 1}\right) \]
Using the identity \( 2\cos^2\theta - 1 = \cos(2\theta) \), we can rewrite \( y \):
\[ y = \sec^{-1}\left(\frac{1}{\cos(2\theta)}\right) \]

### Step 3: Simplify \( y \)
Since \( \sec^{-1}(x) \) is the angle whose secant is \( x \), we have:
\[ y = 2\theta \]

### Step 4: Substitute back for \( \theta \)
Now substituting back for \( \theta \):
\[ y = 2\cos^{-1}(x) \]

### Step 5: Differentiate \( y \) with respect to \( x \)
To find \( \frac{dy}{dx} \), we differentiate \( y \):
\[ \frac{dy}{dx} = 2 \cdot \frac{d}{dx}(\cos^{-1}(x)) \]
Using the derivative of \( \cos^{-1}(x) \):
\[ \frac{d}{dx}(\cos^{-1}(x)) = -\frac{1}{\sqrt{1 - x^2}} \]
Thus,
\[ \frac{dy}{dx} = 2 \cdot \left(-\frac{1}{\sqrt{1 - x^2}}\right) = -\frac{2}{\sqrt{1 - x^2}} \]

### Final Answer
\[ \frac{dy}{dx} = -\frac{2}{\sqrt{1 - x^2}} \]

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