If `f(x) + f(y) = f(xy-sqrt(1-x^(2))sqrt(1-y^(2))), f(0) = (pi)/(2)` and f is differentiable in (-1, 1).
Then `|lim_(x rarr 0)(2f(x)-pi)/(x)|=`

Explanation

To solve the problem step-by-step, we will follow the reasoning laid out in the video transcript while providing a clear mathematical approach.

### Step 1: Understand the given functional equation
We are given the functional equation:
\[ f(x) + f(y) = f(xy - \sqrt{1 - x^2} \sqrt{1 - y^2}) \]
and the condition that \( f(0) = \frac{\pi}{2} \).

### Step 2: Identify the function
From the context and the properties of the function, we suspect that \( f(x) \) could be related to the inverse cosine function. We will assume:
\[ f(x) = \cos^{-1}(x) \]
This assumption is based on the known properties of the cosine function and the functional equation provided.

### Step 3: Verify the function
We need to check if this function satisfies the functional equation:
- Calculate \( f(0) \):
\[ f(0) = \cos^{-1}(0) = \frac{\pi}{2} \]
- Check if the functional equation holds:
\[ f(x) + f(y) = \cos^{-1}(x) + \cos^{-1}(y) \]
The right-hand side:
\[ f(xy - \sqrt{1 - x^2} \sqrt{1 - y^2}) = \cos^{-1}(xy - \sqrt{1 - x^2} \sqrt{1 - y^2}) \]
By the cosine addition formula, we can verify that both sides are equal.

### Step 4: Find the limit
We need to evaluate:
\[ \lim_{x \to 0} \frac{2f(x) - \pi}{x} \]
Substituting \( f(x) = \cos^{-1}(x) \):
\[ \lim_{x \to 0} \frac{2\cos^{-1}(x) - \pi}{x} \]

### Step 5: Evaluate the limit
When \( x = 0 \):
\[ 2\cos^{-1}(0) - \pi = 2 \cdot \frac{\pi}{2} - \pi = 0 \]
This gives us the indeterminate form \( \frac{0}{0} \), so we apply L'Hôpital's Rule.

### Step 6: Apply L'Hôpital's Rule
Differentiate the numerator and denominator:
- The derivative of \( 2\cos^{-1}(x) \) is:
\[ \frac{d}{dx}[2\cos^{-1}(x)] = -\frac{2}{\sqrt{1 - x^2}} \]
- The derivative of \( \pi \) is \( 0 \).
- The derivative of \( x \) is \( 1 \).

Thus, we have:
\[ \lim_{x \to 0} \frac{-\frac{2}{\sqrt{1 - x^2}}}{1} \]

### Step 7: Evaluate the limit as \( x \to 0 \)
Substituting \( x = 0 \):
\[ \lim_{x \to 0} -\frac{2}{\sqrt{1 - 0^2}} = -2 \]

### Step 8: Take the absolute value
Finally, we need to find the absolute value:
\[ | -2 | = 2 \]

### Conclusion
Thus, the final answer is:
\[ \boxed{2} \]