Differentiate `sec^-1""(1)/(2x^2-1)` with respect to `sqrt(1-x^2)`

Options

A. -4
B. 4
C. 2
D. -2

Explanation

To differentiate \( y = \sec^{-1}\left(\frac{1}{2x^2 - 1}\right) \) with respect to \( z = \sqrt{1 - x^2} \), we will follow these steps:

### Step 1: Differentiate \( y \) with respect to \( x \)

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

Using the chain rule, the derivative of \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{1}{\sqrt{\left(\frac{1}{2x^2 - 1}\right)^2 - 1}} \cdot \frac{d}{dx}\left(\frac{1}{2x^2 - 1}\right) \]

Now, we need to differentiate \( \frac{1}{2x^2 - 1} \):
\[ \frac{d}{dx}\left(\frac{1}{2x^2 - 1}\right) = -\frac{2x}{(2x^2 - 1)^2} \]

Thus, substituting back, we have:
\[ \frac{dy}{dx} = \frac{-2x}{(2x^2 - 1)^2 \sqrt{\left(\frac{1}{2x^2 - 1}\right)^2 - 1}} \]

### Step 2: Simplify \( \sqrt{\left(\frac{1}{2x^2 - 1}\right)^2 - 1} \)

We simplify:
\[ \left(\frac{1}{2x^2 - 1}\right)^2 - 1 = \frac{1 - (2x^2 - 1)^2}{(2x^2 - 1)^2} \]
Calculating \( 1 - (2x^2 - 1)^2 \):
\[ = 1 - (4x^4 - 4x^2 + 1) = 4x^2 - 4x^4 \]
Thus,
\[ \sqrt{\left(\frac{1}{2x^2 - 1}\right)^2 - 1} = \frac{\sqrt{4x^2(1 - x^2)}}{2x^2 - 1} \]

### Step 3: Substitute back into \( \frac{dy}{dx} \)

Now substituting this back into our expression for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{-2x}{(2x^2 - 1)^2} \cdot \frac{(2x^2 - 1)}{\sqrt{4x^2(1 - x^2)}} \]
This simplifies to:
\[ \frac{dy}{dx} = \frac{-2x}{(2x^2 - 1) \sqrt{4x^2(1 - x^2)}} \]

### Step 4: Differentiate \( z \) with respect to \( x \)

Next, we differentiate \( z = \sqrt{1 - x^2} \):
\[ \frac{dz}{dx} = \frac{-x}{\sqrt{1 - x^2}} \]

### Step 5: Find \( \frac{dy}{dz} \)

Using the chain rule:
\[ \frac{dy}{dz} = \frac{dy}{dx} \cdot \frac{dx}{dz} \]
Thus,
\[ \frac{dy}{dz} = \frac{dy}{dx} \cdot \frac{1}{\frac{dz}{dx}} = \frac{dy}{dx} \cdot \left(-\frac{\sqrt{1 - x^2}}{x}\right) \]

### Step 6: Substitute \( \frac{dy}{dx} \) into \( \frac{dy}{dz} \)

Substituting \( \frac{dy}{dx} \):
\[ \frac{dy}{dz} = \frac{-2x}{(2x^2 - 1) \sqrt{4x^2(1 - x^2)}} \cdot \left(-\frac{\sqrt{1 - x^2}}{x}\right) \]
This simplifies to:
\[ \frac{dy}{dz} = \frac{2\sqrt{1 - x^2}}{(2x^2 - 1) \sqrt{4x^2}} \]

### Step 7: Evaluate at \( x = \frac{1}{2} \)

Finally, we can evaluate \( \frac{dy}{dz} \) at \( x = \frac{1}{2} \):
\[ \frac{dy}{dz} \bigg|_{x = \frac{1}{2}} = \frac{2\sqrt{1 - \left(\frac{1}{2}\right)^2}}{(2\left(\frac{1}{2}\right)^2 - 1) \sqrt{4\left(\frac{1}{2}\right)^2}} \]
Calculating this gives:
\[ = \frac{2\sqrt{1 - \frac{1}{4}}}{(2 \cdot \frac{1}{4} - 1) \sqrt{4 \cdot \frac{1}{4}}} = \frac{2\sqrt{\frac{3}{4}}}{(0) \cdot 1} \text{ (undefined)} \]