To find the derivative of the function $y = x^2 \ln(x^2)$, we will need to apply the product rule and the chain rule.

The function is a product of two functions:

• $f(x) = x^2$
• $g(x) = \ln(x^2)$

Step 1: Apply the product rule

The product rule states that: $\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$ We need to find the derivatives of $f(x)$ and $g(x)$.

Step 2: Differentiate $f(x) = x^2$

The derivative of $f(x) = x^2$ is: $f'(x) = 2x$

Step 3: Differentiate $g(x) = \ln(x^2)$

Using the chain rule, the derivative of $g(x) = \ln(x^2)$ is: $g'(x) = \frac{d}{dx} \ln(x^2) = \frac{1}{x^2} \cdot 2x = \frac{2}{x}$

Step 4: Apply the product rule

Now that we have $f'(x) = 2x$ and $g'(x) = \frac{2}{x}$, we can apply the product rule: $\frac{d}{dx}[x^2 \ln(x^2)] = (2x) \cdot \ln(x^2) + x^2 \cdot \frac{2}{x}$

Step 5: Simplify the expression

Simplifying the second term: $x^2 \cdot \frac{2}{x} = 2x$ Thus, the derivative is: $\frac{d}{dx}[x^2 \ln(x^2)] = 2x \ln(x^2) + 2x$

Final Answer:

$\frac{d}{dx}[x^2 \ln(x^2)] = 2x (\ln(x^2) + 1)$

Would you like further explanation on any of these steps? Or do you have any questions?

Here are some related questions you might find interesting:

1. How do you apply the chain rule to logarithmic functions?
2. Can you find the second derivative of $y = x^2 \ln(x^2)$?
3. How do you handle derivatives of logarithmic functions with powers in the argument?
4. What is the derivative of $\ln(x)$ with respect to $x$?
5. How do you differentiate functions involving trigonometric and logarithmic combinations?

Tip: When differentiating products, always consider using the product rule, and for logarithms, use the chain rule carefully!