Basic formulas:
- (sin x)' = cos x
- (cos x)' = −sin x
- (tan x)' = 1/cos²x = sec²x
Combined with the chain rule, these can be extended to expressions such as (sin(g(x)))' = cos(g(x))·g'(x).
Example ① Differentiate the following.
(1) y = sin(3x) (2) y = cos²x (3) y = tan(x²)
Solution:
(1) cos(3x)·3 = 3cos(3x)
(2) y = (cos x)². 2cos x·(−sin x) = −2sin x cos x = −sin 2x
(3) (1/cos²(x²))·2x = 2x/cos²(x²)
(1) y = sin(3x) (2) y = cos²x (3) y = tan(x²)
Solution:
(1) cos(3x)·3 = 3cos(3x)
(2) y = (cos x)². 2cos x·(−sin x) = −2sin x cos x = −sin 2x
(3) (1/cos²(x²))·2x = 2x/cos²(x²)
Key Point
- The sin↔cos relationship: sin' = cos, cos' = −sin (watch the minus sign)
- Angles must always be measured in radians
Practice Problems
- Find y' for y = sin(2x+π/3)
- Find y' for y = x·cos x
Answers and Explanations
- Answer:2cos(2x+π/3)
- Answer:cos x − x sin x
Product rule.