The derivative of `cos^(-1)(2x^(2)-1)` w.r.t. `cos^(-1)x` is

Options

A. `2`
B. `(-1)/(2sqrt(1-x^(2)))`
C. `2/x`
D. `1-x^(2)`

Explanation

To find the derivative of \( \cos^{-1}(2x^2 - 1) \) with respect to \( \cos^{-1}(x) \), we will follow these steps:

### Step 1: Define the Functions
Let:
- \( u = \cos^{-1}(2x^2 - 1) \)
- \( v = \cos^{-1}(x) \)

We need to find \( \frac{du}{dv} \).

### Step 2: Find \( \frac{dv}{dx} \)
Using the differentiation formula for \( \cos^{-1}(x) \):
\[ \frac{dv}{dx} = -\frac{1}{\sqrt{1 - x^2}} \]
This is our Equation (1).

### Step 3: Find \( \frac{du}{dx} \)
To differentiate \( u = \cos^{-1}(2x^2 - 1) \), we apply the chain rule:
\[ \frac{du}{dx} = -\frac{1}{\sqrt{1 - (2x^2 - 1)^2}} \cdot \frac{d}{dx}(2x^2 - 1) \]
Calculating \( \frac{d}{dx}(2x^2 - 1) \):
\[ \frac{d}{dx}(2x^2 - 1) = 4x \]
Thus, we have:
\[ \frac{du}{dx} = -\frac{4x}{\sqrt{1 - (2x^2 - 1)^2}} \]

### Step 4: Simplify \( 1 - (2x^2 - 1)^2 \)
We need to simplify \( 1 - (2x^2 - 1)^2 \):
\[ (2x^2 - 1)^2 = 4x^4 - 4x^2 + 1 \]
Thus,
\[ 1 - (2x^2 - 1)^2 = 1 - (4x^4 - 4x^2 + 1) = -4x^4 + 4x^2 = 4x^2(1 - x^2) \]
So, we can rewrite \( \frac{du}{dx} \):
\[ \frac{du}{dx} = -\frac{4x}{\sqrt{4x^2(1 - x^2)}} = -\frac{4x}{2\sqrt{x^2(1 - x^2)}} = -\frac{2}{\sqrt{1 - x^2}} \]
This is our Equation (2).

### Step 5: Find \( \frac{du}{dv} \)
Using the chain rule:
\[ \frac{du}{dv} = \frac{du/dx}{dv/dx} \]
Substituting the values from Equations (1) and (2):
\[ \frac{du}{dv} = \frac{-\frac{2}{\sqrt{1 - x^2}}}{-\frac{1}{\sqrt{1 - x^2}}} \]
This simplifies to:
\[ \frac{du}{dv} = 2 \]

### Final Answer
Thus, the derivative of \( \cos^{-1}(2x^2 - 1) \) with respect to \( \cos^{-1}(x) \) is:
\[ \frac{du}{dv} = 2 \]