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

Explanation

To solve the given problem step by step, we start with the function provided and apply the necessary transformations and differentiation.

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

### Step 2: Substitute \( x \)
Let’s substitute \( x \) with \( \cos \theta \):
\[ x = \cos \theta \]
Then, we can express \( y \) in terms of \( \theta \):
\[ y = \sec^{-1}\left(\frac{1}{2\cos^2\theta - 1}\right) \]

### Step 3: Simplify the expression
Using the double angle identity \( 2\cos^2\theta - 1 = \cos 2\theta \), we can rewrite \( y \):
\[ y = \sec^{-1}\left(\frac{1}{\cos 2\theta}\right) \]

### Step 4: Rewrite using secant
Since \( \frac{1}{\cos 2\theta} = \sec 2\theta \), we have:
\[ y = \sec^{-1}(\sec 2\theta) \]

### Step 5: Simplify further
The secant and its inverse cancel out, leading to:
\[ y = 2\theta \]

### Step 6: Substitute back for \( \theta \)
Recall that \( \theta = \cos^{-1}(x) \). Thus, we can substitute back:
\[ y = 2\cos^{-1}(x) \]

### Step 7: Differentiate \( y \) with respect to \( x \)
Now, we differentiate \( y \):
\[ \frac{dy}{dx} = \frac{d}{dx}[2\cos^{-1}(x)] \]
Using the derivative of \( \cos^{-1}(x) \):
\[ \frac{dy}{dx} = 2 \cdot \left(-\frac{1}{\sqrt{1 - x^2}}\right) \]
So,
\[ \frac{dy}{dx} = -\frac{2}{\sqrt{1 - x^2}} \]

### Final Result
Thus, the derivative is:
\[ \frac{dy}{dx} = -\frac{2}{\sqrt{1 - x^2}} \]

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