If ` y = log |x| , then (dy)/(dx)` =

Options

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

Answer

To find the derivative of the function \( y = \log |x| \), we will consider two cases based on the definition of the absolute value function.

### Step-by-step Solution:

1. **Understanding the Function**:
The function \( y = \log |x| \) can be expressed in two cases based on the value of \( x \):
- Case 1: When \( x > 0 \), \( |x| = x \) and hence \( y = \log x \).
- Case 2: When \( x < 0 \), \( |x| = -x \) and hence \( y = \log (-x) \).

2. **Differentiating Case 1**:
For \( x > 0 \):
\[ y = \log x \]
The derivative of \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{1}{x} \]

3. **Differentiating Case 2**:
For \( x < 0 \):
\[ y = \log (-x) \]
To differentiate this, we apply the chain rule. The derivative of \( \log u \) where \( u = -x \) is:
\[ \frac{dy}{dx} = \frac{1}{u} \cdot \frac{du}{dx} = \frac{1}{-x} \cdot (-1) = \frac{1}{-x} \cdot (-1) = \frac{1}{x} \]

4. **Combining Results**:
From both cases, we can conclude that:
\[ \frac{dy}{dx} = \frac{1}{x} \quad \text{for } x \neq 0 \]

### Final Answer:
Thus, the derivative of \( y = \log |x| \) is:
\[ \frac{dy}{dx} = \frac{1}{x} \quad \text{for } x \neq 0 \]