`y=sec^(-1)""(1)/(2x^(2)-1),0ltxlt(1)/(sqrt(2))`

Explanation

To solve the problem \( y = \sec^{-1}\left(\frac{1}{2x^2 - 1}\right) \) for \( 0 < x < \frac{1}{\sqrt{2}} \), we will differentiate \( y \) with respect to \( x \).

### Step-by-Step Solution

**Step 1: Rewrite the function**
We start with the given function:
\[ y = \sec^{-1}\left(\frac{1}{2x^2 - 1}\right) \]

**Step 2: Set \( x = \cos \theta \)**
To simplify the expression, we can set \( x = \cos \theta \). This gives us:
\[ y = \sec^{-1}\left(\frac{1}{2\cos^2 \theta - 1}\right) \]

**Step 3: Use the double angle identity**
Recall the double angle identity for cosine:
\[ \cos(2\theta) = 2\cos^2(\theta) - 1 \]
Thus, we can rewrite the expression:
\[ y = \sec^{-1}(\sec(2\theta)) \]

**Step 4: Simplify \( y \)**
Since \( y = \sec^{-1}(\sec(2\theta)) \), we have:
\[ y = 2\theta \]

**Step 5: Express \( \theta \) in terms of \( x \)**
From our substitution \( x = \cos \theta \), we can express \( \theta \):
\[ \theta = \cos^{-1}(x) \]
Therefore:
\[ y = 2\cos^{-1}(x) \]

**Step 6: Differentiate \( y \) with respect to \( x \)**
Now 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:
\[ \frac{dy}{dx} = 2 \left(-\frac{1}{\sqrt{1 - x^2}}\right) = -\frac{2}{\sqrt{1 - x^2}} \]

### Final Result
The derivative of \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = -\frac{2}{\sqrt{1 - x^2}} \]