If the primitive of `(1)/(f(x))` is equal to `log{f(x)}^(2)+C`, then f(x) is

Options

A. `x+d`
B. `(x)/(2)+d`
C. `(x^(2))/(2)+d`
D. `x^(2)+d`

Explanation

To solve the problem step by step, we start with the given information:

**Given:**
The primitive (or integral) of \( \frac{1}{f(x)} \) is equal to \( \log{f(x)}^2 + C \).

### Step 1: Understand the given equation
We are given that:
\[ \int \frac{1}{f(x)} \, dx = \log{f(x)}^2 + C \]
This implies that the derivative of \( \log{f(x)}^2 + C \) should equal \( \frac{1}{f(x)} \).

### Step 2: Differentiate the right-hand side
Using the chain rule, we differentiate \( \log{f(x)}^2 \):
\[ \frac{d}{dx}(\log{f(x)}^2) = \frac{d}{dx}(2\log{f(x)}) = 2 \cdot \frac{1}{f(x)} \cdot f'(x) \]
Thus, we have:
\[ \frac{1}{f(x)} = 2 \cdot \frac{f'(x)}{f(x)} \]

### Step 3: Rearranging the equation
From the equation \( \frac{1}{f(x)} = 2 \cdot \frac{f'(x)}{f(x)} \), we can multiply both sides by \( f(x) \):
\[ 1 = 2f'(x) \]

### Step 4: Solve for \( f'(x) \)
From the equation \( 1 = 2f'(x) \), we can isolate \( f'(x) \):
\[ f'(x) = \frac{1}{2} \]

### Step 5: Integrate to find \( f(x) \)
Now we integrate \( f'(x) \) to find \( f(x) \):
\[ f(x) = \int \frac{1}{2} \, dx = \frac{1}{2}x + C \]
Where \( C \) is the constant of integration.

### Conclusion
Thus, the function \( f(x) \) is:
\[ f(x) = \frac{1}{2}x + C \]