- Difficulty
- Standard
- Suggested time
- 3 min
- Marks
- 2
Exponential and trigonometric functions practice question 16
Let . Find .
is times a bracket that also changes with . Which rule does that need, and what is the derivative of each part, signs included?
Product rule with and .
The derivative of is . Collect the terms at the end.
A formula in x. Use ^ for powers and brackets for every input: sin(2x), e^(x^2), log(x+1). Give exact values: 1/6, not 0.167.
Theory UBC Math 100 §4.2CLP-1 §2.8Key results
Worked solution
Strategy
Use the product rule, then collect: the cosine terms cancel.
Working
u′v + uv′; the derivative of sin x − cos x is cos x + sin x.
Factor out eˣ; the cosines cancel.
Answer
The answer is f prime of x equals 2 e to the x sin x.
Check
The formula 2eˣ sin x is positive on (0, π), and f does climb there, from f(0) = −1 to f(π) = eπ ≈ 23.14.
Where this goes wrong
Differentiating −cos x as −sin x. The bracket then differentiates to cos x − sin x, everything cancels, and f′ comes out as 0, which cannot be right for a function that is not constant.
Leans on
The product rule, from Differentiation rules
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