If `y = sec^(-1) ((1)/(2x^(2) -1)) " then " (dy)/(dx)`= ?

Options

A. `(-2)/((1 + x^(2)))`
B. `(-2)/((1 -x^(2)))`
C. `(-2)/(sqrt(1 -x^(2)))`
D. none of these

Explanation

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

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

### Step 2: Use the Property of Inverse Secant
Recall that if \( y = \sec^{-1}(u) \), then:
\[ \sec(y) = u \]
Thus, we can write:
\[ \sec(y) = \frac{1}{2x^2 - 1} \]

### Step 3: Express in Terms of Cosine
Using the identity \( \sec(y) = \frac{1}{\cos(y)} \), we can rewrite the equation as:
\[ \frac{1}{\cos(y)} = \frac{1}{2x^2 - 1} \]
This implies:
\[ \cos(y) = 2x^2 - 1 \]

### Step 4: Differentiate Both Sides
Now we differentiate both sides with respect to \( x \). Using implicit differentiation:
\[ -\sin(y) \frac{dy}{dx} = 4x \]
Here, we used the chain rule on the left side.

### Step 5: Solve for \( \frac{dy}{dx} \)
Rearranging the equation gives us:
\[ \frac{dy}{dx} = -\frac{4x}{\sin(y)} \]

### Step 6: Express \( \sin(y) \) in Terms of \( x \)
From the identity \( \sin^2(y) + \cos^2(y) = 1 \), we can find \( \sin(y) \):
\[ \sin^2(y) = 1 - \cos^2(y) = 1 - (2x^2 - 1)^2 \]
Calculating \( (2x^2 - 1)^2 \):
\[ (2x^2 - 1)^2 = 4x^4 - 4x^2 + 1 \]
Thus,
\[ \sin^2(y) = 1 - (4x^4 - 4x^2 + 1) = -4x^4 + 4x^2 \]
So,
\[ \sin(y) = \sqrt{4x^2(1 - x^2)} = 2x\sqrt{1 - x^2} \]

### Step 7: Substitute Back into the Derivative
Substituting \( \sin(y) \) back into the derivative:
\[ \frac{dy}{dx} = -\frac{4x}{2x\sqrt{1 - x^2}} = -\frac{2}{\sqrt{1 - x^2}} \]

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

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