`(logx)/(x)`

Answer

To differentiate the function \( y = \frac{\log x}{x} \), we will use the quotient rule. The quotient rule states that if you have a function in the form \( \frac{u}{v} \), then its derivative is given by:

\[ \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

where \( u = \log x \) and \( v = x \).

### Step 1: Identify \( u \) and \( v \)
Let:
- \( u = \log x \)
- \( v = x \)

### Step 2: Differentiate \( u \) and \( v \)
Now we need to find \( \frac{du}{dx} \) and \( \frac{dv}{dx} \):
- The derivative of \( u \) is:
\[ \frac{du}{dx} = \frac{1}{x} \]
- The derivative of \( v \) is:
\[ \frac{dv}{dx} = 1 \]

### Step 3: Apply the Quotient Rule
Now we can apply the quotient rule:
\[ \frac{dy}{dx} = \frac{x \cdot \frac{1}{x} - \log x \cdot 1}{x^2} \]

### Step 4: Simplify the Expression
Now simplify the expression:
\[ \frac{dy}{dx} = \frac{1 - \log x}{x^2} \]

### Final Result
Thus, the derivative of \( y = \frac{\log x}{x} \) is:
\[ \frac{dy}{dx} = \frac{1 - \log x}{x^2} \]

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