- Difficulty
- Standard
- Suggested time
- 5 min
- Marks
- 3
The definition of the derivative practice question 10
Use the definition of the derivative to find for the function below.
The difference quotient here has a fraction inside a fraction. How do you tidy that up before letting ?
Write with and .
Combine the two fractions in the numerator over a common denominator before dividing by ; a factor of will cancel.
A fraction, or a decimal to two decimal places.
Theory UBC Math 100 §3.3CLP-1 §2.2Key results
Worked solution
Strategy
Set up the difference quotient, combine the fractions so that the h in the numerator becomes visible, cancel it, and then let h → 0.
Working
f(1 + h) = 3/(1 + h + 2) and f(1) = 3/3 = 1.
Common denominator. The numerator is a multiple of h, as it must be.
Cancel h, then substitute h = 0.
Answer
The answer is negative one third.
Check
f(1.01) = 3/3.01 ≈ 0.99668, so over a step of 0.01 the function falls about 0.00332, a slope near −0.332.
Where this goes wrong
Setting h = 0 before cancelling, which gives 0/0, or losing the minus sign when subtracting (3 + h) from 3. The numerator is −h, and the answer is negative because f is decreasing.
Leans on
Combining fractions over a common denominator, from Functions, graphs and inverses
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