- Difficulty
- Standard
- Suggested time
- 3 min
- Marks
- 2
Differentiation rules practice question 7
Let . Find .
h is two brackets multiplied together, and both change with x. Which rule handles a product, and is it just the product of the two derivatives?
Use the product rule: .
Take and , so and .
A whole number.
Theory UBC Math 100 §4.1CLP-1 §2.4Key results
Worked solution
Strategy
This is a product of two polynomials. The product rule handles it directly; expanding first also works and is a useful check.
Working
Name the factors and differentiate each one before combining.
Apply u′v + uv′. Neither term is optional.
Expand and collect like terms.
Substitute the point into the simplified derivative.
Answer
The answer is 22.
Check
h(2) = 5 and h(2.01) ≈ 5.2213, a rise of about 0.22 over 0.01, so a slope near 22.
Where this goes wrong
Writing h′ = u′v′ = 2 · 2x = 4x, which gives 8. The derivative of a product is not the product of the derivatives; both cross terms are needed.
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