- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 1
The chain rule and logarithms practice question 3
Let for . Find .
Look inside the logarithm: it holds , not . Could a log law tidy that up before you differentiate?
Either use the chain rule, , or simplify first: for .
A fraction, or a decimal to two decimal places.
Theory UBC Math 100 §4.4CLP-1 §2.10Key results
Worked solution
Strategy
Both routes work and agree. The log law route is shorter; the chain rule route confirms it.
Working
Log of a power brings the exponent down (valid because x > 0).
u′/u with u = x².
Substitute.
Answer
The answer is two thirds.
Check
Raising x from 3 to 3.03 (1%) raises x² by about 2%, so log(x²) rises by about 0.02 over a step of 0.03, a slope of about 2/3.
Where this goes wrong
Writing the derivative of log(x²) as 1/x², forgetting the inner derivative 2x. The chain rule applies to logarithms just as it does to powers.
Leans on
Laws of logarithms, from Functions, graphs and inverses
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