- Topic
- Antiderivatives
- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
- Theory
- CLP-1 §4.1Key results
Antiderivatives practice question 3
A function is defined for and satisfies there. Which of the following gives every such ?
Two tests for each choice: is it defined for every , and does it differentiate to ?
and need . Differentiate each remaining candidate, using the chain rule where there is an inner function, and compare with .
Theory CLP-1 §4.1Key results
Worked solution
Strategy
Test each candidate twice: is it defined for every x < 0, and does it differentiate to 1/x? The chain rule settles the second.
Working
Chain rule; the two minus signs cancel.
The full family on the negative axis.
Answer
The answer is natural log of negative x, plus a constant.
Check
From x = −2 to −1.99, log(−x) changes by log 0.995 ≈ −0.0050, a slope of about −0.5, which is 1/x at x = −2.
Where this goes wrong
Writing −log x + C to “fix” the sign. That is undefined for x < 0 and differentiates to −1/x anyway. The sign goes inside the logarithm, not in front of it.
Leans on
The derivative of log, from The chain rule and logarithms
Try a similar question
A function is defined for all and satisfies for every , with and . Find .Saved in this browser only.