Calculating from the definition every time is difficult, so we use formulas.
Basic formulas：
- (x^n)′ = n·x^{n−1}（valid even when n is a real number）
- (c)′ = 0（the derivative of a constant is 0）
- {c·f(x)}′ = c·f'(x)
- {f(x) ± g(x)}′ = f'(x) ± g'(x)
Example ① Differentiate the following.
(1) y = x³ − 2x² + 5x − 7 (2) y = √x (3) y = 1/x²
Solution:
(1) y' = 3x² − 4x + 5
(2) y = x^{1/2} → y' = (1/2)·x^{−1/2} = 1/(2√x)
(3) y = x^{−2} → y' = −2·x^{−3} = −2/x³
(1) y = x³ − 2x² + 5x − 7 (2) y = √x (3) y = 1/x²
Solution:
(1) y' = 3x² − 4x + 5
(2) y = x^{1/2} → y' = (1/2)·x^{−1/2} = 1/(2√x)
(3) y = x^{−2} → y' = −2·x^{−3} = −2/x³
Key Points
- (x^n)′ = nx^{n−1} is valid even when n is fractional or negative
- Rewrite √x and 1/x^k using exponents to apply the formula
Practice Problems
- Find y' for y = 2x⁴ − 3x² + x.
- Find y' for y = 3√x + 1/x.
Answers and Explanations
- Answer:8x³ − 6x + 1
- Answer:3/(2√x) − 1/x²
Differentiate 3x^{1/2} + x^{−1}。