- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 2
Exponential and trigonometric functions practice question 5
Let for . Find .
Sort the three terms by where the variable sits: in the base, in the exponent, or nowhere. Remember that is a constant, about 2.718.
takes the power rule, is its own derivative, and is a number.
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 §3.5CLP-1 §2.7Key results
Worked solution
Strategy
All three terms are built from e and x, but each needs a different rule. Decide for each term whether x is the base, the exponent, or absent.
Working
Variable base, constant exponent: the power rule works for any constant exponent.
Constant base e, variable exponent: eˣ is its own derivative.
No x at all: eᵉ is a constant, about 15.15.
Add the three derivatives.
Answer
The answer is f prime of x equals e times x to the power e minus 1, plus e to the x.
Check
At x = 1 the formula gives 2e ≈ 5.44, and (f(1.01) − f(0.99))/0.02 ≈ 5.44; the constant eᵉ ≈ 15.15 adds nothing to either.
Where this goes wrong
Keeping eᵉ in the derivative, or differentiating it as e · ee−1, which is eᵉ again. It contains no x, so like any number its derivative is 0.
Leans on
The power rule, from Differentiation rules
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