To find the derivative of the function $y = x^2 \ln(x^2)$, let's go through the process step-by-step.

Step 1: Rewrite the Function

Since $\ln(x^2)$ can be simplified, we can rewrite the function as: $y = x^2 \cdot \ln(x^2) = x^2 \cdot 2 \ln(x) = 2x^2 \ln(x)$

So now, we have $y = 2x^2 \ln(x)$.

Step 2: Differentiate Using the Product Rule

The product rule states that for two functions $u(x)$ and $v(x)$, the derivative $(uv)' = u'v + uv'$.

Here, we let:

• $u = 2x^2$
• $v = \ln(x)$

Differentiate $u$ and $v$:

1. $u = 2x^2$, so $u' = 4x$.
2. $v = \ln(x)$, so $v' = \frac{1}{x}$.

Apply the Product Rule:

Using the product rule: $y' = u'v + uv'$ Substitute $u, u', v,$ and $v'$: $y' = (4x) \ln(x) + (2x^2) \cdot \frac{1}{x}$

Step 3: Simplify the Expression

$y' = 4x \ln(x) + 2x$

Final Answer

The derivative of $y = x^2 \ln(x^2)$ is: $y' = 4x \ln(x) + 2x$

Would you like more details on any of the steps, or have any questions?

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Here are five related questions for further practice:

1. Find the derivative of $y = x^3 \ln(x)$.
2. Determine the derivative of $y = x^2 \ln(x^3)$.
3. If $y = x^2 \ln(x) + x$, find $y'$.
4. Evaluate the derivative of $y = x^2 e^{\ln(x)}$.
5. Find the second derivative of $y = 2x^2 \ln(x)$.

Tip: When dealing with logarithmic expressions, simplify any exponents inside the logarithm before differentiating.