- Difficulty
- Standard
- Suggested time
- 5 min
- Marks
- 3
The chain rule and logarithms practice question 14
Use logarithmic differentiation to find for the function below.
The question asks for logarithmic differentiation. What does become once the log laws split the product, the quotient and the powers?
Take of both sides and expand with the log laws: , valid for .
Differentiate: . Evaluate at 2, then multiply by .
A fraction, or a decimal to two decimal 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 > 1.
Differentiate both sides; the left side is (log f)′ = f′/f by the chain rule.
Evaluate at x = 2.
Multiply by f(2).
Answer
The answer is 76 ninths, about 8.444.
Check
f(1.99) ≈ 2.583 and f(2.01) ≈ 2.752, a rise of about 0.169 over 0.02, a slope near 8.44.
Where this goes wrong
Stopping at f′/f = 19/6. Logarithmic differentiation gives the derivative of log f; multiply by f(2) = 8/3 to get f′(2).
Leans on
Laws of logarithms, from Functions, graphs and inverses
Try a similar question
Let . Use logarithmic differentiation to find .Saved in this browser only.