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AP Calculus · Unit 2 · Differentiation: definition and basic rules · last updated

Differentiation: definition and basic rules practice questions (AP Calculus)

Unit 2 covers the derivative as a limit and as the slope of the tangent line, differentiability and continuity, the power, product and quotient rules, and the derivatives of the basic trigonometric, exponential and logarithmic functions. Below are all 2 free AP Calculus questions on it from our free exams, each with a worked answer behind a toggle, then 6 key terms with the formulas typeset, and a door to make an AP Calculus-style paper on this unit from your own notes. The free exams are typeset PDFs with an answer key; the exam downloads without a signup.

Make a practice exam from your notes →Upload your notesAP Calculus style preselected; tick "Unit 2 · Differentiation: definition and basic rules" on the next screen. Free to try, no card.

Sample questions

All 2 free AP Calculus questions on this unit, from the free AP Calculus AB Practice Exam. Each answer is behind its own toggle; the whole paper, its PDF and the full answer key are on the exam page.

  1. Question 12 marks · Limit definition of the derivative
    lim⁡h→09+h−3h\lim_{h \to 0} \dfrac{\sqrt{9 + h} - 3}{h} is:
    • A.
      00
    • B.
      16\frac{1}{6}
    • C.
      13\frac{1}{3}
    • D.
      nonexistent
    From the free AP Calculus AB Practice Exam →
  2. Question 22 marks · Product rule
    If f(x)=x2exf(x) = x^2 e^x, then f′(x)=f'(x) =
    • A.
      2xex2x e^x
    • B.
      x2exx^2 e^x
    • C.
      ex(x2+2x)e^x(x^2 + 2x)
    • D.
      ex(x2−2x)e^x(x^2 - 2x)
    From the free AP Calculus AB Practice Exam →

Key terms

The definitions and formulas AP Calculus expects for differentiation: definition and basic rules, typeset as they appear on the paper.

Definition of the derivative
The limit of the difference quotient, which is the slope of the tangent line and the instantaneous rate of change. The exam also uses the alternative form at a point.
f′(a)=lim⁡h→0f(a+h)−f(a)h=lim⁡x→af(x)−f(a)x−af'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}
Differentiability and continuity
A function differentiable at a point is continuous there, but not the reverse: ∣x∣|x| is continuous at 0 yet has no derivative there (a corner). Cusps, vertical tangents and discontinuities all break differentiability.
Power rule and constant multiples
The workhorse rule for polynomials, roots and reciprocals once they are written as powers of xx.
ddxxn=nxn−1,ddx(cf(x))=cf′(x)\frac{d}{dx} x^n = n x^{n-1}, \qquad \frac{d}{dx}\big(c f(x)\big) = c f'(x)
Product rule
The derivative of a product is not the product of the derivatives; each factor is differentiated in turn.
(fg)′=f′g+fg′(fg)' = f'g + fg'
Quotient rule
Derivative of the top times the bottom, minus the top times the derivative of the bottom, over the bottom squared. Order matters in the numerator.
(fg)′=f′g−fg′g2\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}
Derivatives of the basic functions
The results assumed on every later unit; the derivative of cos⁡x\cos x carries a minus sign.
ddxsin⁡x=cos⁡x,ddxcos⁡x=−sin⁡x,ddxex=ex,ddxln⁡x=1x\frac{d}{dx}\sin x = \cos x, \quad \frac{d}{dx}\cos x = -\sin x, \quad \frac{d}{dx} e^x = e^x, \quad \frac{d}{dx}\ln x = \frac{1}{x}

Frequently asked questions

Are there free differentiation: definition and basic rules practice questions for AP Calculus?

Yes. 2 questions on differentiation: definition and basic rules appear in the free AP Calculus AB Practice Exam, and the sample questions on this page come from it with worked answers. Each exam can be read online or downloaded as a typeset PDF without signing up, and its full answer key is emailed on request.

How do I make an AP Calculus practice paper on differentiation: definition and basic rules from my notes?

Open the create page with the AP Calculus style preselected, tick "Unit 2 · Differentiation: definition and basic rules" under the units to cover (or upload your notes and slides on it), choose the question mix and time limit, and generate. You get a typeset PDF in the AP format with a separate answer key, in about a minute. It is free to try, no card needed.

What does AP Calculus expect you to know about differentiation: definition and basic rules?

Unit 2 covers the derivative as a limit and as the slope of the tangent line, differentiability and continuity, the power, product and quotient rules, and the derivatives of the basic trigonometric, exponential and logarithmic functions. The terms to have cold: Definition of the derivative, Differentiability and continuity, Power rule and constant multiples, Product rule, Quotient rule. Each is defined below with its formula typeset where one applies.

What is the AP Calculus exam format?

The AP Calculus guide explains the papers, their timing and mark allocation, with a sample question and mark scheme and the common mistakes, and links to the generator with the AP Calculus style preselected.

Other AP Calculus topics

All AP Calculus topics · AP Calculus exam guide · All free practice exams

Make an AP Calculus paper on differentiation: definition and basic rules from your notes

Open "Pick your exam" with the AP Calculus style preselected and tick "Unit 2 · Differentiation: definition and basic rules", or upload your notes on it. You get a typeset paper in the AP format with a separate answer key.

Make a practice exam from your notes →Upload your notes