- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 2
Differentiation rules practice question 23
A student finds for the function below as shown. Find the first line that is wrong.
Both brackets change when x changes. How many terms does the derivative of a product have?
Expand f first and differentiate the polynomial, then compare with line 1.
How sure are you?
Theory UBC Math 100 §4.1CLP-1 §2.4Key results
Corrected working
Strategy
Product rule: differentiate each factor in turn and add. Expanding first gives the same answer.
Working
u′v + uv′.
Substitute x = 1.
Answer
The answer is f prime of 1 equals 0.
Check
f(0.99) and f(1.01) are both about −1.9993, just above f(1) = −2, so f bottoms out at x = 1 and f′(1) = 0 fits.
Where this goes wrong
Differentiating one factor and leaving the other alone. The product rule has two terms because both factors change with x.
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