To find the derivative of the function \( y = \frac{\ln x}{x} \), we will use the quotient rule of differentiation. The quotient rule states that if you have a function in the form \( \frac{u}{v} \), where both \( u \) and \( v \) are functions of \( x \), 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 need to find \( \frac{du}{dx} \) and \( \frac{dv}{dx} \):
1. **Differentiate \( u = \ln x \)**:
\[
\frac{du}{dx} = \frac{1}{x}
\]
2. **Differentiate \( v = x \)**:
\[
\frac{dv}{dx} = 1
\]
Now we can substitute these derivatives into the quotient rule formula:
\[
\frac{dy}{dx} = \frac{x \cdot \frac{1}{x} - \ln x \cdot 1}{x^2}
\]
This simplifies to:
\[
\frac{dy}{dx} = \frac{1 - \ln x}{x^2}
\]
Thus, the derivative of \( y = \frac{\ln x}{x} \) is:
\[
\frac{dy}{dx} = \frac{1 - \ln x}{x^2}
\]
To find the derivative of the function \( y = \frac{\ln x}{x} \), we will use the quotient rule of differentiation. The quotient rule states that if you have a function in the form \( \frac{u}{v} \), where both \( u \) and \( v \) are functions of \( x \), 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:
- \( u = \ln x \)
...
