To find the derivative of the function \( y = \frac{\ln x}{x} \), we will use the quotient rule. The quotient rule states that if you have a function that is the quotient of two functions \( u(x) \) and \( v(x) \), then the derivative is given by:
\[
\frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}
\]
In our case, we can identify:
- \( u = \ln x \)
- \( v = x \)
Now, we will find the derivatives of \( u \) and \( v \):
1. **Differentiate \( u \)**:
\[
\frac{du}{dx} = \frac{d}{dx}(\ln x) = \frac{1}{x}
\]
2. **Differentiate \( v \)**:
\[
\frac{dv}{dx} = \frac{d}{dx}(x) = 1
\]
Now, we can apply the quotient rule:
\[
\frac{dy}{dx} = \frac{x \cdot \frac{1}{x} - \ln x \cdot 1}{x^2}
\]
3. **Substituting the values**:
\[
\frac{dy}{dx} = \frac{x \cdot \frac{1}{x} - \ln x}{x^2}
\]
4. **Simplifying the expression**:
\[
\frac{dy}{dx} = \frac{1 - \ln x}{x^2}
\]
Thus, the derivative \( \frac{dy}{dx} \) is:
\[
\frac{dy}{dx} = \frac{1 - \ln x}{x^2}
\]
### Final Answer:
\[
\frac{dy}{dx} = \frac{1 - \ln x}{x^2}
\]
To find the derivative of the function \( y = \frac{\ln x}{x} \), we will use the quotient rule. The quotient rule states that if you have a function that is the quotient of two functions \( u(x) \) and \( v(x) \), then the derivative is given by:
\[
\frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}
\]
In our case, we can identify:
- \( u = \ln x \)
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