Differentiate `cos^(-1)(1-2x^2)` with respect to `x ,` if `0

Explanation

To differentiate the function \( f(x) = \cos^{-1}(1 - 2x^2) \) with respect to \( x \), we will apply the chain rule of differentiation. The chain rule states that if you have a composite function \( f(g(x)) \), then the derivative is given by \( f'(g(x)) \cdot g'(x) \).

### Step-by-Step Solution:

1. **Identify the outer and inner functions**:
- Outer function: \( f(u) = \cos^{-1}(u) \) where \( u = 1 - 2x^2 \)
- Inner function: \( g(x) = 1 - 2x^2 \)

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