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The first derivative of \( y = x^2 \cos(2x) \) is:

\[
\frac{dy}{dx} = 2x(\cos(2x) - x \sin(2x))
\]

From the provided options, the correct answer is:

\[
\boxed{2x (\cos 2x - x \sin 2x)}
\]

Do you need further details or have any questions?

Here are 5 related questions to expand your understanding:
1. What is the second derivative of \( y = x^2 \cos(2x) \)?
2. How does the product rule apply to the function \( y = x^2 \cos(2x) \)?
3. Can you find the derivative of \( y = x^2 \sin(2x) \) using a similar method?
4. How would the derivative change if the function were \( y = x^3 \cos(2x) \)?
5. What is the derivative of \( y = e^x \cos(2x) \)?

**Tip**: Always check if the product rule or chain rule is required when differentiating a product of functions or composite functions.

What is the first derivative of y = x^2 cos 2x?

Find the first derivative of the function y = x^2 cos(2x) using the product rule and trigonometric differentiation techniques. This problem explores key calculus concepts including the product rule and chain rule. Suitable for students in Grades 11-12 or College Calculus.

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