- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 3
Differentiation rules practice question 11
The table gives values of two differentiable functions and and of their derivatives. Let . Find .
No formula for or is given, only a table. Which four entries does the rule for a quotient need at ?
Quotient rule: . The numerator starts with the derivative of the top.
Use the row : , , , . Square for the denominator.
An integer.
Theory UBC Math 100 §4.1CLP-1 §2.6Key results
Worked solution
Strategy
No formula for f or g is needed: the quotient rule uses only the values and derivatives at the point, which is exactly what the table supplies.
Working
The quotient rule at x = 3.
Substitute from the row x = 3.
Simplify. The denominator is positive because it is a square.
Answer
The answer is negative 6.
Check
Q(3) = −2, and near x = 3 the top grows while the negative bottom moves toward 0, so Q becomes more negative and Q′(3) must be negative.
Where this goes wrong
Reversing the numerator to f g′ − f′g, which gives +6. The quotient rule starts with the derivative of the top: f′g − f g′.
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