If f(x) `=|log_(e)|x||,` then f'(x) equals

Options

A. `(1)/(|x|)` , where `x ne 0`
B. `(1)/(x)` for `|x| gt 1` and `-(1)/(x)` for `|x| lt 1`
C. `-(1)/(x)` for `|x| gt 1` and `(1)/(x)` for `|x| lt 1`
D. `(1)/(x)` for `x gt 0` and `- (1)/(x)` for `x lt 0`

Explanation

To find the derivative \( f'(x) \) of the function \( f(x) = |\log_e |x|| \), we will analyze the function in different intervals based on the value of \( x \).

### Step 1: Analyze the function based on intervals

The function \( f(x) = |\log_e |x|| \) can be split into different cases based on the value of \( x \):

1. **Case 1:** \( x > 1 \)
2. **Case 2:** \( 0 < x \leq 1 \)
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