- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 3
The definition of the derivative practice question 9
Use the limit definition of the derivative to find for .
Rules such as the power rule earn no marks on this question.
What two values of does the definition need when ? Work each one out in full before any limit is taken.
Write in full before simplifying. Every occurrence of becomes , including the one inside .
After subtracting , every term left in the numerator should carry a factor of you can cancel.
A single number.
Theory UBC Math 100 §3.3CLP-1 §2.2Key results
Worked solution
Strategy
Substitute into the difference quotient, expand carefully, simplify until the h in the denominator cancels, and only then let h approach 0.
Working
Replace every x with 3 + h, including inside the linear term.
Expand and collect like terms.
Evaluate the function at the point itself.
Subtract, then cancel. The h in the denominator is gone, so the limit is now routine.
Let h approach 0. The tangent at x = 3 rises 2 units for every unit to the right.
Answer
The answer is 2.
Check
A small step gives f(3.01) − f(3) = 0.0201, a slope of 2.01 over 0.01, close to 2.
Where this goes wrong
Expanding (3 + h)² correctly but writing −4(3 + h) as −12, or as −12 + 4h. The −4 multiplies both terms of the bracket, giving −12 − 4h; distributing over the whole bracket is where this question is usually lost.
Leans on
Expanding binomials, from Functions, graphs and inverses
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