Study guides › Level 10 · Calculus

Derivatives of polynomials

Level 10 · Calculus · for questions like “f(x) = 2x³ + 4x² − 5x + 1. What is f′(x)?”

💡 The idea

Do each term on its own with the power rule: bring the power down, then lower it by one. Numbers on their own disappear.

✏️ Worked examples

1. f(x) = 2x³ + 4x² − 5x + 1: 2x³ → 6x², 4x² → 8x, −5x → −5, and 1 → 0. So f′(x) = 6x² + 8x − 5.

2. f(x) = x³ − 7x: f′(x) = 3x² − 7.

⚠️ Watch out

• Lower the power: x³ → 3x², not 3x³.
• Keep the signs: −5x → −5.

🎯 Try these

1. f(x) = 4x³ + x²
2. f(x) = 5x² − 3x + 8
3. f(x) = x³ − 6x² + 9x

Answers: 1) 12x² + 2x 2) 10x − 3 3) 3x² − 12x + 9

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