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.