Recognising derivative over function
A logarithm appears when the numerator is a constant multiple of the derivative of the denominator.
Find the constant match
Match the numerator to the denominator derivative
1 · name it
bottom = f(x)
2 · differentiate
top = kf′(x)
3 · reverse
k ln |f(x)|
No constant match? Do not force the log method.
The denominator must be non-zero. On a definite interval, it must not cross zero between the limits.
- Name the denominator .
- Differentiate it and compare with the complete numerator.
- Keep the constant multiple and the modulus bars.
Examiner note
Know when the match fails
Worked example
A constant multiple is enough
The denominator derivative is , and:
By contrast, is not one logarithm: no constant multiple of produces . Split it into an -part and a constant part.
Key idea
Common mistake
(Pure Mathematics 2 & 3 Coursebook, Ch. 8.3)
Recognise tangent as the same pattern
The denominator derivative is , so the required constant multiple is .
Common mistake
Find , showing how the numerator matches the denominator derivative.