A

Step 1: We start with the given function \( y = \frac{\ln x}{x} \).
Step 2: To find the derivative \( \frac{dy}{dx} \), we will apply the quotient rule: \( \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2} \), where \( u = \ln x \) and \( v = x \).
Step 3: We find the derivatives: \( u' = \frac{1}{x} \) and \( v' = 1 \).
Step 4: Applying the quotient rule, we have:
\( \frac{dy}{dx} = \frac{\left(\frac{1}{x}\right)x - \ln x \cdot 1}{x^2} = \frac{1 - \ln x}{x^2} \).
Step 5: This gives us the final result of the derivative. Therefore, the answer is A.