Differentiation: definition and fundamental properties practice problems AP Calculus, Unit 2
The derivative as a limit, the power, product and quotient rules, and derivatives of sin, cos, ex and ln x. Try each question before you look: use the hint if you are stuck, then reveal the worked solution one step at a time.
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Key equations for differentiation: definition and fundamental properties
| f'(x) = limh→0 [f(x + h) − f(x)]/h | the definition of the derivative |
|---|---|
| d/dx xn = nxn−1 | power rule |
| (uv)' = u'v + uv' | product rule |
| (u/v)' = (u'v − uv')/v2 | quotient rule |
| (sin x)' = cos x, (ex)' = ex, (ln x)' = 1/x | basic derivatives |
Questions marked No calculator should be done by hand, as in Part A of the multiple-choice section. You can type fractions, such as 3/4. The full sheet is in our AP Calculus formula sheet.
Warm-up questions
Quick checks on the core ideas. If these feel easy, go straight to the exam-style questions.
Hint
Use the power rule.
Worked solution
- f'(x) = 3x2.
- f'(2) = 3 × 4 = 12.
Hint
This is the definition of a derivative. Of which function, at which point?
Worked solution
- It is f'(2) for f(x) = x2, since f(2) = 4.
- f'(x) = 2x, so f'(2) = 4.
- The answer is C.
Hint
Differentiate term by term.
Worked solution
- The derivative is 20x3 − 3.
- At x = 1: 20 − 3 = 17.
Exam-style questions
Questions at the level of the AP Calculus exam. The ones marked Challenge combine more than one idea.
Hint
Use the product rule with u = x2 and v = ex.
Worked solution
- f'(x) = 2x ex + x2ex.
- f'(1) = 2e + e = 3e.
- 3e = 8.155.
Hint
You need the point and the slope at x = 1.
Worked solution
- Point: y(1) = 1 − 2 = −1, so (1, −1).
- Slope: y' = 3x2 − 2, so y'(1) = 1.
- Tangent: y + 1 = 1(x − 1), so y = x − 2. The y-intercept is −2.
Hint
Use the quotient rule.
Worked solution
- f'(x) = [(1)(x − 1) − (x + 1)(1)]/(x − 1)2 = −2/(x − 1)2.
- f'(2) = −2/1 = −2.
Hint
Look at the slope from each side.
Worked solution
- The graph is unbroken at 0, so f is continuous.
- The slope is −1 from the left and +1 from the right: a corner, so no derivative.
- The answer is B. Differentiable always implies continuous, but not the other way round.
Hint
Product rule, or rewrite as ½ sin 2x.
Worked solution
- d/dx(sin x cos x) = cos2x − sin2x = cos 2x.
- At x = 0: cos 0 = 1.
Free-response practice
One original question in the style of the exam's free-response section. Answer it in full, with working, before you open the scoring guide.
No-calculator free-response style · 5 points · about 15 minutes
Let f(x) = x2 − 4x.
- (a) Use the limit definition of the derivative to find f'(x).
- (b) Find the equation of the tangent line at x = 3.
- (c) Find the x-coordinate where the tangent line is horizontal.
Write your full answer on paper first, then open the scoring guide and tick each point you earned.
Scoring guide and model answer
Part (a)
[f(x + h) − f(x)]/h = [2xh + h2 − 4h]/h = 2x + h − 4 → 2x − 4 as h → 0.
Part (b)
f(3) = −3 and f'(3) = 2, so y + 3 = 2(x − 3), or y = 2x − 9.
Part (c)
f'(x) = 2x − 4 = 0 at x = 2.
Your mark: 0 of 5
Common mistakes in differentiation: definition and fundamental properties
- Multiplying derivatives instead of using the product rule.
- Mixing up the order of the terms in the quotient rule numerator.
- Thinking a continuous function must be differentiable.
Keep going
Revise the ideas behind these questions:
Then try real exam questions: AP Calculus past papers, with official scoring guidelines. · All AP Calculus units