In which interval of x the function `lnx/x` is decreasing?

Answer

To determine the interval of \( x \) for which the function \( f(x) = \frac{\ln x}{x} \) is decreasing, we will follow these steps:

### Step 1: Find the derivative of the function
We start with the function:
\[ f(x) = \frac{\ln x}{x} \]
To find where this function is decreasing, we first need to compute its derivative \( f'(x) \).

Using the quotient rule:
\[ f'(x) = \frac{(x \cdot \frac{d}{dx}(\ln x) - \ln x \cdot \frac{d}{dx}(x))}{x^2} \]
Calculating the derivatives:
\[ \frac{d}{dx}(\ln x) = \frac{1}{x}, \quad \frac{d}{dx}(x) = 1 \]
Substituting these into the derivative:
\[ f'(x) = \frac{x \cdot \frac{1}{x} - \ln x \cdot 1}{x^2} = \frac{1 - \ln x}{x^2} \]

### Step 2: Set the derivative less than zero
To find where the function is decreasing, we set the derivative less than zero:
\[ f'(x) < 0 \implies \frac{1 - \ln x}{x^2} < 0 \]
Since \( x^2 > 0 \) for all \( x > 0 \), we can focus on the numerator:
\[ 1 - \ln x < 0 \implies 1 < \ln x \]

### Step 3: Solve the inequality
To solve the inequality \( 1 < \ln x \), we exponentiate both sides:
\[ e^1 < x \implies e < x \]
Thus, we conclude that:
\[ x > e \]

### Step 4: State the interval
The function \( f(x) = \frac{\ln x}{x} \) is decreasing for:
\[ x \in (e, \infty) \]

### Final Answer
The function \( \frac{\ln x}{x} \) is decreasing in the interval \( (e, \infty) \).