- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 3
Differentiation rules practice question 9
Let . Find .
is a quotient. Before applying the rule, look closely at the bottom, : what is its derivative, sign included?
Quotient rule with and . Note that , which flips a sign in the numerator.
Differentiate in general first, then substitute , minding every sign.
An exact value, or a decimal to three places.
Theory UBC Math 100 §4.1CLP-1 §2.6Key results
Worked solution
Strategy
Apply the quotient rule carefully, keeping the derivative of the denominator (which is −1) explicit, and only then substitute x = −1.
Working
Name the parts.
The −1 turns the subtraction into an addition; then expand and collect.
Substitute. The denominator is 3² = 9.
Answer
The answer is negative ten ninths, about −1.111.
Check
f(−0.99) ≈ 2.6556 and f(−1.01) ≈ 2.6778, so f falls about 0.022 over a run of 0.02, a slope near −1.11.
Where this goes wrong
Treating the derivative of 2 − x as +1, which gives (−18 − 8)/9 = −26/9. The denominator decreases as x increases, so v′ = −1.
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