Learn the differentiation formulas for polynomials so that you can calculate derivatives quickly.
Basic Formulas
- (c)' = 0 (differentiation of a constant)
- (xⁿ)' = nxⁿ⁻¹
- (cf(x))' = cf'(x)
- (f(x)+g(x))' = f'(x)+g'(x)
Example ①
Differentiate the following: (1) y = 3x⁴ − 2x² + 5x − 1 (2) y = x³ − 4x
Answer:
(1) y' = 12x³ − 4x + 5
(2) y' = 3x² − 4
Differentiate the following: (1) y = 3x⁴ − 2x² + 5x − 1 (2) y = x³ − 4x
Answer:
(1) y' = 12x³ − 4x + 5
(2) y' = 3x² − 4
Example ②
Expand y = (x+1)(x²−x+1) and then differentiate it.
Answer:y = x³ + 1。y' = 3x²
Expand y = (x+1)(x²−x+1) and then differentiate it.
Answer:y = x³ + 1。y' = 3x²
Key Points
- (xⁿ)' = nxⁿ⁻¹: bring the exponent down and reduce it by 1
- The derivative of a constant term is 0
- Expand products before differentiating (the product rule is also used in Mathematics II)
Practice Problems
- Differentiate the following: (1) y = 5x³ − 3x² + 2 (2) y = x⁴ − 2x³ + x
- Expand y = (2x−1)² and then differentiate it.
- When y = x³, find x such that y' = 0.
Answers and Explanations
- Answer:(1) 15x²−6x (2) 4x³−6x²+1
Explanation: Differentiate each term using the formulas. - Answer:y = 4x²−4x+1 → y' = 8x−4
Explanation:(2x−1)² = 4x²−4x+1。 - Answer:x = 0
Explanation:y' = 3x² = 0 → x = 0。