- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 1
Exponential and trigonometric functions practice question 1
Let . Find .
Two standard derivatives are needed, one for and one for . Can you state each from memory, sign included?
is its own derivative, and the derivative of is . Differentiate term by term, then substitute .
A whole number.
Theory UBC Math 100 §4.2CLP-1 §2.8Key results
Worked solution
Strategy
Both functions have derivatives you should know outright. Differentiate each term with its constant, then evaluate at 0, where e⁰ = 1 and cos 0 = 1.
Working
Derivative of eˣ is eˣ; derivative of sin x is cos x.
e⁰ = 1 and cos 0 = 1.
Answer
The answer is 2.
Check
Near 0, eˣ ≈ 1 + x and sin x ≈ x, so f(x) ≈ 4 + 2x, a line of slope 2.
Where this goes wrong
Writing the derivative of sin x as −cos x. It is cos x; the minus sign belongs to the derivative of cos x.
Leans on
Values of eˣ, sin and cos at 0, from Functions, graphs and inverses
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