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The text in the image asks you to describe the steps for simplifying the following expression:

\[
\frac{d}{dx} \left(\frac{x^2}{\ln(x)}\right)
\]

Here’s a step-by-step explanation for simplifying this derivative:

### Step-by-step solution:

1. **Recognize the quotient rule**: The given expression is a quotient of two functions \( x^2 \) and \( \ln(x) \). To differentiate this, we apply the **quotient rule**, which states:

\[
\frac{d}{dx} \left(\frac{u}{v}\right) = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2}
\]

Where:
- \( u = x^2 \)
- \( v = \ln(x) \)

2. **Differentiate the numerator** \( u = x^2 \):
\[
\frac{du}{dx} = 2x
\]

3. **Differentiate the denominator** \( v = \ln(x) \):
\[
\frac{dv}{dx} = \frac{1}{x}
\]

4. **Apply the quotient rule**:
\[
\frac{d}{dx} \left(\frac{x^2}{\ln(x)}\right) = \frac{\ln(x) \cdot 2x - x^2 \cdot \frac{1}{x}}{\ln^2(x)}
\]

5. **Simplify the expression**:
The second term simplifies as:
\[
x^2 \cdot \frac{1}{x} = x
\]
So, the derivative becomes:
\[
\frac{2x \ln(x) - x}{\ln^2(x)}
\]

Thus, the simplified form of the derivative is:
\[
\frac{2x \ln(x) - x}{\ln^2(x)}
\]

---

Would you like any further clarifications or additional details? Here are five related questions that might help deepen your understanding:

1. How is the quotient rule derived, and when should it be applied?
2. What happens if the function in the denominator does not have a well-defined derivative at certain points?
3. Can this method be applied to functions involving trigonometric or exponential terms?
4. How would you differentiate a product of two functions using the product rule instead of the quotient rule?
5. What are some common mistakes to avoid when applying the quotient rule?

**Tip**: Always simplify both the numerator and the denominator as much as possible before applying the quotient rule to make the process smoother.

Describe, in words, your first step in simplifying d/dx (x^2 / ln(x)). Then finish simplifying the expression. Write the answer in radical form (if necessary). Check your final answer.

Learn how to differentiate x^2 / ln(x) by applying the quotient rule. Follow the detailed steps to simplify the expression, including derivatives of polynomial and logarithmic functions. Suitable for students in advanced high school or college-level calculus.

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