- Difficulty
- Challenging
- Suggested time
- 5 min
- Marks
- 3
Differentiation rules practice question 21
The table gives values at of two twice-differentiable functions and and of their first and second derivatives. Let . Find .
The table goes up to second derivatives, so you need in terms of , and their derivatives. Which rule gets you there, and how many times?
Product rule once: . Each of those two terms is itself a product.
Differentiate each of those products with the product rule again, collect like terms, then read the table.
A number.
Theory UBC Math 100 §3.4CLP-1 §2.14Key results
Worked solution
Strategy
Apply the product rule twice. The two middle terms are equal, which is where the factor 2 in h″ = f″g + 2f′g′ + fg″ comes from.
Working
The product rule once.
The product rule on each term; the two f′g′ terms combine.
Substitute from the table.
Answer
The answer is negative 15.
Check
Quadratics with the table’s values, f = 3 − u + 2u² and g = −2 + 5u + ½u² (u = x − 2), multiply to a u² coefficient of −7.5, half of −15.
Where this goes wrong
Writing h″ = f″g + f′g′ + fg″ with a single middle term, which gives −10. Each of f′g and fg′ produces an f′g′ term, so there are two.
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