To find the derivative of \( x^2 \times \ln\left(\frac{2}{x}\right) \), we will apply the product rule and chain rule.

### Step-by-step solution:

We have a product of two functions:

\[
f(x) = x^2, \quad g(x) = \ln\left(\frac{2}{x}\right)
\]

The derivative of a product \( f(x) \times g(x) \) is given by the product rule:

\[
\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
\]

1. **Derivative of** \( f(x) = x^2 \):

\[
f'(x) = 2x
\]

2. **Derivative of** \( g(x) = \ln\left(\frac{2}{x}\right) \):

We can rewrite \( g(x) \) as:

\[
g(x) = \ln(2) - \ln(x)
\]

Now, differentiate it:

\[
g'(x) = -\frac{1}{x}
\]

3. **Apply the product rule**:

Now substitute into the product rule:

\[
\frac{d}{dx} \left( x^2 \times \ln\left(\frac{2}{x}\right) \right) = 2x \ln\left(\frac{2}{x}\right) + x^2 \times \left(-\frac{1}{x}\right)
\]

Simplify the expression:

\[
= 2x \ln\left(\frac{2}{x}\right) - x
\]

Thus, the derivative of \( x^2 \times \ln\left(\frac{2}{x}\right) \) is:

\[
\boxed{2x \ln\left(\frac{2}{x}\right) - x}
\]

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How would you find the second derivative of this function?
2. What happens to the derivative as \( x \) approaches zero?
3. Can you find the critical points of the original function using the first derivative?
4. How would the derivative change if the function were \( x^3 \times \ln\left(\frac{2}{x}\right) \)?
5. How can you apply logarithmic differentiation to this problem?

**Tip:** When differentiating a product, always break it down into the product rule, and carefully apply it to avoid mistakes in handling complex expressions.

Find the derivative of x^2 × ln(2/x) by applying the product and chain rules. This problem involves calculus concepts such as logarithmic differentiation and the product rule, suitable for students in Grades 11-12 and early university.

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