Find $\frac{dy}{dx}$. If y = 3x3 + 2x2 + 3x + 1

Analyze the question: We need to differentiate the polynomial y = 3x³ + 2x² + 3x + 1 with respect to x. (The "3x3" and "2x2" in the question are 3x³ and 2x².)

Concept: Use the power rule:

d/dx (axⁿ) = n·a·xⁿ⁻¹

The derivative of a constant is 0, and we differentiate each term separately.

Solution:

• d/dx (3x³) = 3 × 3 × x² = 9x²
• d/dx (2x²) = 2 × 2 × x = 4x
• d/dx (3x) = 3
• d/dx (1) = 0

Adding these together:

dy/dx = 9x² + 4x + 3

Tip: For each term, multiply the coefficient by the power, then reduce the power by 1. Remember that a plain number (constant) disappears when you differentiate, and a term like 3x just becomes 3. Watch the signs carefully, since the other options only differ by their signs.