Differentiate y = 4x³ − 2x + 7.

## The reasoning We're using the **power rule of differentiation**: bring down the exponent as a coefficient, then reduce the exponent by 1. For y = 4x³ − 2x + 7: - **4x³** becomes 4 × 3x² = **12x²** - **−2x** (which is −2x¹) becomes −2 × 1x⁰ = **−2** - **7** (a constant) disappears because constants have zero rate of change So dy/dx = **12x² − 2** ## Why the wrong options tempt you **Option B (12x² + 2)**: You forgot the negative sign on −2x. Always carry signs through your working! **Option C (4x² − 2)**: You forgot to multiply by the original exponent (3). The power rule says "exponent comes down as a multiplier" — don't skip that step. **Option D (x³ − 2x)**: You didn't differentiate at all; you just rewrote part of the original function. Differentiation *changes* the expression. ## Quick takeaway **Power rule: multiply by the power, then subtract 1 from the power. Constants vanish.** Practice until it's automatic!