We solve problems by combining the formulas studied so far（basic, product, and quotient rules）.
Example ① Find y' for y = (2x+1)/√x.
Solution:y = (2x+1)·x^{−1/2} = 2x^{1/2} + x^{−1/2}
y' = x^{−1/2} − (1/2)x^{−3/2} = 1/√x − 1/(2x√x)
Solution:y = (2x+1)·x^{−1/2} = 2x^{1/2} + x^{−1/2}
y' = x^{−1/2} − (1/2)x^{−3/2} = 1/√x − 1/(2x√x)
Example ② Find y' for y = x²(x−1)³（combining the product rule and basic formulas）.
Solution:(x²)'·(x−1)³ + x²·{(x−1)³}'
= 2x(x−1)³ + x²·3(x−1)²·1 = (x−1)²·{2x(x−1) + 3x²} = (x−1)²(5x²−2x) = x(x−1)²(5x−2)
Solution:(x²)'·(x−1)³ + x²·{(x−1)³}'
= 2x(x−1)³ + x²·3(x−1)²·1 = (x−1)²·{2x(x−1) + 3x²} = (x−1)²(5x²−2x) = x(x−1)²(5x−2)
Key Points
- When you see the form: fraction→exponential notation, composition→the chain rule studied next
- Factor out common factors to organize the calculation
Practice Problems
- Find y' for y = (x²+1)/(x+1)
- Find y' for y = x³(2x−1)
Answers and Explanations
- Answer:(x²+2x−1)/(x+1)²
(2x(x+1)−(x²+1))/(x+1)² = (x²+2x−1)/(x+1)²。 - Answer:8x³ − 3x²
Expand to 2x⁴−x³ and differentiate。