If `f(x)=log|x|,xne0` then f'(x) equals

Options

A. `1/(|x|)`
B. `1/x`
C. `-1/x`
D. ` +- 1/x`

Answer

To find the derivative of the function \( f(x) = \log |x| \) where \( x \neq 0 \), we can follow these steps:

### Step 1: Rewrite the function
The function \( f(x) = \log |x| \) can be expressed in piecewise form:
- For \( x > 0 \): \( f(x) = \log x \)
- For \( x < 0 \): \( f(x) = \log (-x) \)

### Step 2: Differentiate for \( x > 0 \)
For \( x > 0 \):
\[ f'(x) = \frac{d}{dx} (\log x) = \frac{1}{x} \]

### Step 3: Differentiate for \( x < 0 \)
For \( x < 0 \):
\[ f'(x) = \frac{d}{dx} (\log (-x)) = \frac{1}{-x} \cdot (-1) = \frac{1}{x} \]

### Step 4: Combine the results
From the differentiation results, we have:
- For \( x > 0 \), \( f'(x) = \frac{1}{x} \)
- For \( x < 0 \), \( f'(x) = \frac{1}{x} \)

Thus, we can write:
\[ f'(x) = \frac{1}{x} \quad \text{for } x \neq 0 \]

### Final Answer
Therefore, the derivative \( f'(x) \) is:
\[ f'(x) = \frac{1}{x} \quad \text{for } x \neq 0 \]

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