- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 3
The chain rule and logarithms practice question 10
Let . Find at .
This is a composition: an inner quadratic raised to a fifth power. Use the chain rule.
Differentiate the outer power first, leaving the inside untouched: . Then multiply by the derivative of the inside, and put in only at the end.
A whole number.
Theory UBC Math 100 §4.3CLP-1 §2.9Key results
Worked solution
Strategy
Differentiate the outside function with the inside left alone, then multiply by the derivative of the inside. Substitute only at the end.
Working
Outer derivative first. The bracket is carried through unchanged.
The inner derivative is 6x. Multiply the constants.
Evaluate the inside at x = 1 before raising to the fourth power.
Substitute and compute. 4⁴ = 256.
Answer
The answer is 7680.
Check
y(0.999) ≈ 1016.35 and y(1.001) ≈ 1031.71, a rise of about 15.36 over a step of 0.002, a slope of 7680.
Where this goes wrong
Stopping at 5(3x² + 1)⁴ and forgetting the inner derivative, which gives 1280. The chain rule is a product of two derivatives, and the second one is easy to leave out.
Leans on
The power rule, from Differentiation rules
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