- Difficulty
- Standard
- Suggested time
- 5 min
- Marks
- 3
The chain rule and logarithms practice question 20
Use logarithmic differentiation to find for the function below.
Every factor of is a power, the square root included. Which log law handles a power, and what power is ?
Take logs and expand ( is the natural logarithm): , valid for .
Differentiate to get , evaluate it at 1, then multiply by .
An exact value, or a decimal to three places.
Theory UBC Math 100 §4.4CLP-1 §2.10Key results
Worked solution
Strategy
A product and quotient of powers becomes a sum of simple logarithms once log is taken. Differentiate the sum, then multiply by f to undo the logarithm.
Working
Products add, quotients subtract, powers come down. Every factor is positive for x > 0.
Differentiate both sides; the left side is (log f)′ = f′/f by the chain rule.
Evaluate at x = 1.
Multiply by f(1).
Answer
The answer is one twelfth, about 0.083.
Check
f rises from about 0.07324 at x = 0.99 to 0.07491 at x = 1.01, a slope of about 0.083.
Where this goes wrong
Stopping at f′/f = 9/8. Logarithmic differentiation gives the derivative of log f; multiply by f(1) = 2/27 to get f′(1).
Leans on
Laws of logarithms, from Functions, graphs and inverses
Try a similar question
Use logarithmic differentiation to find for the function below.Saved in this browser only.