Application problems combining composite, trigonometric, exponential, and logarithmic functions.
Example ① Find y' for y = e^x · sin x.
Solution: Product rule. y' = e^x sin x + e^x cos x = e^x(sin x + cos x)
Solution: Product rule. y' = e^x sin x + e^x cos x = e^x(sin x + cos x)
Example ② Find y' for y = log(cos x).
Solution: (1/cos x)·(−sin x) = −tan x
Solution: (1/cos x)·(−sin x) = −tan x
Key Point
- Differentiate from the outside inward in order (chain rule)
- For products and quotients, distinguish the functions before calculating
Practice Problems
- Find y' for y = sin(e^x)
- Find y' for y = e^{2x}·cos(3x)
Answers and Explanations
- Answer:e^x · cos(e^x)
- Answer:e^{2x}{2cos(3x) − 3sin(3x)}