If `f(x)=log|x|,xne0`, then what is `f'(x)` equal to ?

Options

A. `(1)(|x|)`
B. `(1)/(x)`
C. `(-1)/(x)`
D. None of these

Explanation

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

### Step 1: Understand the function
The function is defined as \( f(x) = \log|x| \). The absolute value is important because it allows the logarithm to be defined for both positive and negative values of \( x \), except for zero.

### Step 2: Differentiate the function
To differentiate \( f(x) = \log|x| \), we can use the property of logarithms:
- For \( x > 0 \), \( \log|x| = \log x \)
- For \( x < 0 \), \( \log|x| = \log(-x) \)

Thus, we can differentiate \( f(x) \) separately for positive and negative \( x \).

### Step 3: Differentiate for \( x > 0 \)
For \( x > 0 \):
\[ f(x) = \log x \]
The derivative is:
\[ f'(x) = \frac{1}{x} \]

### Step 4: Differentiate for \( x < 0 \)
For \( x < 0 \):
\[ f(x) = \log(-x) \]
The derivative is:
\[ f'(x) = \frac{1}{-x} \cdot (-1) = \frac{1}{x} \]

### Step 5: Combine the results
Since both cases yield the same derivative:
\[ f'(x) = \frac{1}{x} \quad \text{for } x \neq 0 \]

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

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