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The problem asks to find the derivative \( dy \) of the function \( y = 2x^2 \).

To solve this, we need to compute the derivative of \( y \) with respect to \( x \):

\[
y = 2x^2
\]

Using the power rule for differentiation, which states \( \frac{d}{dx} [x^n] = n x^{n-1} \), we differentiate \( y \):

\[
\frac{dy}{dx} = 2 \cdot \frac{d}{dx} [x^2] = 2 \cdot 2x = 4x
\]

Thus, the derivative \( \frac{dy}{dx} = 4x \), and multiplying by \( dx \) gives:

\[
dy = 4x \, dx
\]

Therefore, the correct answer is **a. \( 4x \, dx \)**.

Would you like further details or have any questions?

Here are five related questions to deepen understanding:
1. How is the power rule for differentiation applied in other cases?
2. What happens when the coefficient in front of \( x^2 \) is changed?
3. How does the chain rule affect differentiation in composite functions?
4. What are the interpretations of \( dy \) and \( dx \) in real-life scenarios?
5. How does implicit differentiation differ from explicit differentiation?

**Tip**: Always apply the power rule carefully when differentiating powers of \( x \).

Learn how to differentiate the function y = 2x^2 using the power rule. This detailed solution explains the application of differentiation and covers key formulas like dy/dx = nx^(n-1). Ideal for students in Grades 11-12.

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