- Difficulty
- Standard
- Suggested time
- 5 min
- Marks
- 3
The chain rule and logarithms practice question 21
Let . Find .
Three factors would mean the product rule twice. Is there a way to turn this product of powers into a sum before you differentiate?
A product of three powers: take logs ( is the natural logarithm) to turn it into a sum, .
Differentiate each logarithm with the chain rule, evaluate at 1, then multiply by .
An exact value.
Theory UBC Math 100 §4.4CLP-1 §2.10Key results
Worked solution
Strategy
Three factors would need the product rule twice. Logarithms turn the product into a sum of three terms, each differentiated by the chain rule, and multiplying by f at the end undoes the logarithm.
Working
Products add and powers come down. Every factor is positive for x > −1.
Each logarithm of a function differentiates to the inner derivative over the function.
Evaluate at x = 1.
Multiply by f(1).
Answer
The answer is 448.
Check
f(0.999) ≈ 63.554 and f(1.001) ≈ 64.450, a rise of about 0.896 over a step of 0.002, a slope of 448.
Where this goes wrong
Stopping at f′/f = 7. That is the derivative of log f; f′(1) is 7 times f(1) = 64.
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.