Let `f(x)=log_(e)|x-1|, x ne 1`, then the value of `f'((1)/(2))` is

Options

A. `-2`
B. 2
C. non-existent
D. 1

Explanation

To find the derivative \( f'(\frac{1}{2}) \) for the function \( f(x) = \log_e |x - 1| \) where \( x \neq 1 \), we can follow these steps:

### Step 1: Define the function in piecewise form
The function \( f(x) = \log_e |x - 1| \) can be expressed in two parts based on the value of \( x \):
- For \( x > 1 \): \( f(x) = \log_e (x - 1) \)
- For \( x < 1 \): \( f(x) = \log_e (1 - x) \)

### Step 2: Find the derivative of \( f(x) \)
Now we will differentiate \( f(x) \) for both cases.

1. **For \( x > 1 \)**:
\[ f'(x) = \frac{d}{dx} \log_e (x - 1) = \frac{1}{x - 1} \]

2. **For \( x < 1 \)**:
\[ f'(x) = \frac{d}{dx} \log_e (1 - x) = \frac{-1}{1 - x} = \frac{1}{x - 1} \]

Thus, the derivative \( f'(x) \) can be summarized as:
\[ f'(x) = \frac{1}{x - 1} \quad \text{for all } x \neq 1 \]

### Step 3: Evaluate \( f'(\frac{1}{2}) \)
Since \( \frac{1}{2} < 1 \), we use the derivative formula:
\[ f'(\frac{1}{2}) = \frac{1}{\frac{1}{2} - 1} = \frac{1}{-\frac{1}{2}} = -2 \]

### Conclusion
The value of \( f'(\frac{1}{2}) \) is \( -2 \).