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The image contains mathematical content that appears to involve calculus or differential equations. Let me analyze the image to extract the relevant mathematical expressions and provide a solution or explanation.

Please give me a moment to read and interpret the content.The image contains a mathematical problem involving a differential equation. Here is the extracted content from the image:

Given:
\[
f(x) = \ln(x) \quad \text{and} \quad g(x) = x^2
\]

Compute the following derivative:
\[
\frac{d}{dx} \left( \frac{f(x)}{g(x)} \right)
\]

### Solution

The derivative of a quotient can be computed using the Quotient Rule, which states:

\[
\frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}
\]

Here, \( u(x) = f(x) = \ln(x) \) and \( v(x) = g(x) = x^2 \).

1. Compute the derivatives:
   - \( u'(x) = \frac{d}{dx}[\ln(x)] = \frac{1}{x} \)
   - \( v'(x) = \frac{d}{dx}[x^2] = 2x \)

2. Apply the Quotient Rule:
\[
\frac{d}{dx} \left( \frac{\ln(x)}{x^2} \right) = \frac{\left( \frac{1}{x} \right)(x^2) - (\ln(x))(2x)}{(x^2)^2}
\]

3. Simplify the expression:
\[
= \frac{x - 2x \ln(x)}{x^4}
\]

4. Further simplify:
\[
= \frac{x(1 - 2\ln(x))}{x^4}
\]
\[
= \frac{1 - 2\ln(x)}{x^3}
\]

### Final Answer
\[
\frac{d}{dx} \left( \frac{\ln(x)}{x^2} \right) = \frac{1 - 2\ln(x)}{x^3}
\]

Would you like more details on this solution, or do you have any questions?

Here are 5 related questions that you might find useful:

1. How do you find the derivative of a product using the product rule?
2. What is the derivative of \( \ln(x) \) with respect to \( x \)?
3. How can you find the derivative of a function raised to a power?
4. What are some practical applications of the quotient rule in calculus?
5. How do the rules for differentiation change when working with higher-order derivatives?

**Tip:** Always keep the rules for differentiation handy, such as the product rule, quotient rule, and chain rule, as they help solve various types of problems in calculus efficiently.

The image contains a mathematical problem involving a differential equation.

Solve the calculus problem involving the derivative of ln(x) over x^2 using the Quotient Rule. This detailed step-by-step solution explains the key concepts, including the use of the Quotient Rule, and is suitable for college-level calculus students.

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