If `f(x)=|logx|`, then for `xne1,f'(x)` equals

Options

A. `1//x`
B. `1//|x|`
C. `-1//x`
D. none of these

Explanation

To solve the problem, we need to differentiate the function \( f(x) = | \log x | \) for \( x \neq 1 \).

### Step 1: Determine the expression for \( f(x) \)

The function \( f(x) = | \log x | \) can be expressed in piecewise form based on the value of \( \log x \):

- For \( x < 1 \): \( \log x < 0 \) so \( f(x) = -\log x \)
- For \( x > 1 \): \( \log x > 0 \) so \( f(x) = \log x \)

### Step 2: Differentiate \( f(x) \)

Now we differentiate \( f(x) \) in both cases:

1. **For \( x < 1 \)**:
\[ f(x) = -\log x \]
The derivative is:
\[ f'(x) = -\frac{d}{dx}(\log x) = -\frac{1}{x} \]

2. **For \( x > 1 \)**:
\[ f(x) = \log x \]
The derivative is:
\[ f'(x) = \frac{d}{dx}(\log x) = \frac{1}{x} \]

### Step 3: Combine the results

Thus, we have:
- For \( x < 1 \): \( f'(x) = -\frac{1}{x} \)
- For \( x > 1 \): \( f'(x) = \frac{1}{x} \)

### Conclusion

Since the question specifies \( x \neq 1 \), we conclude that:
- For \( x < 1 \), \( f'(x) = -\frac{1}{x} \)
- For \( x > 1 \), \( f'(x) = \frac{1}{x} \)

Since both derivatives are valid for their respective intervals and the options provided do not cover both cases, the correct answer is **none of the above**.