The derivative of the product or quotient of two functions is not as simple as that of a sum or difference.
Product rule：{f·g}′ = f'·g + f·g'
Quotient rule：(f/g)′ = (f'·g − f·g')/g²
Example ① (Product) Find y' for y = (x²+1)(x−3).
Solution:y' = 2x·(x−3) + (x²+1)·1 = 2x²−6x + x²+1 = 3x²−6x+1
Solution:y' = 2x·(x−3) + (x²+1)·1 = 2x²−6x + x²+1 = 3x²−6x+1
Example ② (Quotient) Find y' for y = x/(x²+1).
Solution:y' = (1·(x²+1) − x·2x)/(x²+1)² = (1−x²)/(x²+1)²
Solution:y' = (1·(x²+1) − x·2x)/(x²+1)² = (1−x²)/(x²+1)²
Key Points
- Product: remember “first·second' + first'·second”
- Quotient: denominator is g², numerator is f'·g − f·g'（pay attention to the order）
Practice Problems
- Find y' for y = (3x−1)(x²+2)
- Find y' for y = (2x+1)/(x−3)
Answers and Explanations
- Answer:9x² − 2x + 6
3(x²+2) + (3x−1)·2x = 3x²+6 + 6x²−2x。 - Answer:−7/(x−3)²
(2(x−3) − (2x+1))/(x−3)² = −7/(x−3)²。