- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 1
Exponential and trigonometric functions practice question 2
Which of the following is the derivative of ?
Where does the variable sit in : in the base or in the exponent? That decides which rules can apply.
The power rule is for a variable base and a constant exponent. Here the exponent is the variable. Write .
Theory UBC Math 100 §3.5CLP-1 §2.7Key results
Worked solution
Strategy
An exponential with base a is e raised to (log a)·x. Differentiating that brings down the constant log a.
Working
Rewrite with base e.
The derivative of ekx is k·ekx, with k = log 3.
Answer
The answer is 3 to the x times the natural log of 3.
Check
At x = 0, (30.001 − 1)/0.001 ≈ 1.099, which is log 3 ≈ 1.0986, not the 1 that 3ˣ alone would give.
Where this goes wrong
Applying the power rule to get x·3x−1. That rule needs a constant exponent; when the variable is in the exponent the derivative picks up a factor of log 3 instead.
Leans on
Rewriting aˣ as ex log a, from Functions, graphs and inverses
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