- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 1
The chain rule and logarithms practice question 1
Let . Find .
The fourth power sits on the whole bracket , not on alone. Which rule handles a power of an expression?
Chain rule: differentiate the outer power with the inside untouched, then multiply by the derivative of the inside, which is 2.
A whole number.
Theory UBC Math 100 §4.3CLP-1 §2.9Key results
Worked solution
Strategy
The function is a power of a linear expression. The chain rule gives the outer derivative times the inner derivative; the inner derivative is the constant 2.
Working
Outer derivative 4u³, inner derivative 2.
Substitute x = 0.
Answer
The answer is 8.
Check
Between x = −0.01 and x = 0.01, f goes from 0.98⁴ ≈ 0.9224 to 1.02⁴ ≈ 1.0824, a rise of about 0.16 over 0.02, so a slope of 8.
Where this goes wrong
Forgetting the inner derivative and answering 4. The factor of 2 from differentiating 2x + 1 is the whole point of the chain rule.
Leans on
The power rule, from Differentiation rules
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