- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
Differentiation rules practice question 4
The table gives values of two differentiable functions and and of their derivatives. Let . Find .
Which entries of the table does really need: the values of and , or their derivatives? And what becomes of the ?
Differentiate term by term: and . Then read and from the row .
An integer.
Theory UBC Math 100 §4.1CLP-1 §2.4Key results
Worked solution
Strategy
The derivative of a sum is the sum of the derivatives, and a constant multiple comes along unchanged. So h′(2) needs only f′(2) and g′(2), which the table gives, and the derivative of x³.
Working
Linearity: differentiate each term and keep its constant.
Read the derivative columns in the row x = 2. The values f(2) and g(2) are not needed.
Substitute. The two minus signs in the middle term make it positive.
Answer
The answer is 30.
Check
At x = 2, 3f and x³ alone grow at 12 + 12 = 24, and −2g rises too because g is falling, so h′(2) must exceed 24.
Where this goes wrong
Writing −2 · (−3) as −6, which gives 18. The minus sign in front of 2g and the negative value of g′(2) cancel.
Try a similar question
Let . Find .Saved in this browser only.