AP Calculus AB · College Board CED Unit 2

Differentiation · Definition
& Basic Rules

Where limits become something useful. A derivative is just the slope of a tangent line — but that one idea gives you velocity, marginal cost, growth rate, and the entire toolkit of calculus in one breath.

Press → to begin

38 SLIDES · 80 MIN · BUILT FROM CED SKILLS 3.1–3.7

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2 · Learning Objectives

By the end of this unit, you will be able to…

Six measurable goals aligned to College Board CED Topic 3.

LO 1 Distinguish average rate of change from instantaneous rate (the derivative).
LO 2 Evaluate derivatives from the limit definition — without memorized rules.
LO 3 Apply power rule, constant rule, sum/difference rule to polynomials.
LO 4 Differentiate e^x, ln x, sin x, cos x, tan x — and know their geometric origins.
LO 5 Use product rule, quotient rule, and combinations on composite expressions.
LO 6 Identify where a derivative does NOT exist (corners, cusps, vertical tangents, jumps).

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3 · AP CED Mapping

Unit 2 → AP Calculus AB Topic 3

Topic 3 is the single biggest topic on the AP exam. ≈17-20% of MCQ points and shows up in every FRQ.

Topic	CED Skill ID	Weight	Slides
3.1 Derivative at a point	CHA-2.B / C	2-3%	6–10
3.2 Derivative as a function	CHA-2.D	2-3%	11–12
3.3 Basic derivative rules	FUN-3.A	5-7%	13–19
3.4 Trig derivatives	FUN-3.B	2-3%	17–19
3.5 Product & quotient rules	FUN-3.B	5-7%	20–23
3.6 Differentiability	FUN-3.B	1-2%	11, 24
3.7 Higher-order derivatives	CHA-2.D	1-2%	25–26

⚡

If you learn only one unit for AP Calculus AB, learn this one. ~50% of every problem you'll ever see starts with computing or interpreting a derivative.

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4 · Key Terms

Key Terms · Part 1

Six words you'll hear for the next 100 slides. Memorize.

Average rate of change

Δy / Δx = (f(b) − f(a)) / (b − a) — slope of the secant line.

Instantaneous rate

lim as the interval shrinks to one point → slope of tangent line.

Derivative f'(a)

The slope of the tangent to y = f(x) at x = a. Defined as a limit of difference quotients.

Difference quotient

[f(a + h) − f(a)] / h or [f(x) − f(a)] / (x − a).

Differentiable at a

The derivative f'(a) exists as a finite real number — the secant limit exists.

Tangent line

y = f(a) + f'(a) · (x − a). Best linear approximation at a.

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5 · Key Terms (cont.) & Notation

Five ways to write the derivative

Notation	Used by	Most common when…
f'(x)	Lagrange	talking about a function f
dy / dx	Leibniz	talking about variables y and x
d/dx f(x)	Operational	operations on a formula
D_x y	Newton-esque	emphasizing which variable you differentiate with respect to
y'	Prime	when y is the dependent variable

🔁

5 notations, 1 meaning. Pick whichever you can read fastest. The AP exam accepts any of them.

Higher-order derivatives: f'(x) is the first derivative. f''(x) (also written d²y/dx²) is the second derivative. f'''(x) is the third. After that, we use (n): f⁽⁴⁾(x).

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6 · Topic 3.1 · Average vs Instantaneous (graph)

What does "instantaneous rate" actually look like?

Take a curve. Pick two points, draw a secant. Now squeeze the right point toward the left. The secant swings around — and converges to the tangent.

B → A: secant swings toward the tangent. The limit of that swing is the instantaneous slope.

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7 · Topic 3.1 · Definition of f'(a)

The derivative, formally

Two equivalent definitions of f'(a) · CHA-2.B

f'(a) = lim_h→0 [f(a+h) − f(a)] / h

f'(a) = lim_x→a [f(x) − f(a)] / (x − a)

Both are the limit of a slope (difference quotient) as the second point approaches the first.
Either form defines the slope of the tangent line to y = f(x) at x = a.
If that limit exists and is finite, f is differentiable at a.

🫠

Meme break

I had a great joke about the definition of the derivative — but the rate at which it gets funny is approaching zero. (h limit real)

📏

Meme break

A secant line and a tangent line were roommates. The secant kept cutting through the curve twice; the tangent only touched it once. The tangent said: "respect the commitment." (The limit process = moving in and never leaving.)

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8 · Worked · Use the Definition

Find f'(x) from first principles

Q · power rule from scratch

Q. For f(x) = x², use the limit definition to find f'(3).

Step 1 Set up the difference quotient: (f(3+h) − f(3)) / h = ((3+h)² − 9) / h.

Step 2 Expand: (9 + 6h + h² − 9) / h = (6h + h²) / h = 6 + h.

Step 3 Take limit h → 0: f'(3) = 6.

🔁

The result 6 matches the power rule (d/dx x² = 2x, evaluated at x = 3 → 6). The definition is the engine the shortcut formulas came from.

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9 · Worked · Harder Definition

Find f'(x) — square root version

Q · using the conjugate trick from Unit 1

Q. f(x) = √x. Compute f'(a) for a > 0.

Step 1 Difference quotient: (√(a+h) − √a) / h.

Step 2 Multiply top & bottom by conjugate (√(a+h) + √a): ( (a+h) − a ) / ( h(√(a+h) + √a) ) = 1 / (√(a+h) + √a).

Step 3 Take h → 0: f'(a) = 1 / (2√a) = (1/2) a^−1/2.

Power rule confirms it: x^1/2 → (1/2) x^−1/2. ✓ Same answer, just one line. The definition is always there when you need it — but once you trust the rules, you can fly.

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10 · Tangent Line Equation

The tangent line at a point

If you know f(a) and f'(a), you have the tangent line

y = f(a) + f'(a) · (x − a)

Point-slope form. This is the best linear approximation of f near x = a.

Example

Q. f(x) = x³ + 2x at x = 1. Find the tangent line.

→ f(1) = 3. f'(x) = 3x² + 2 ⇒ f'(1) = 5. Tangent: y = 3 + 5(x − 1) = 5x − 2.

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11 · Topic 3.6 · Differentiability implies continuity

Differentiable ⇒ Continuous (not the reverse)

Theorem

If f is differentiable at a, then f is continuous at a.
Converse is false. f(x) = |x| is continuous at 0 but not differentiable there.

🪞

AP trap: "Differentiable on [−1, 1]" is a stricter condition than "Continuous on [−1, 1]." All differentiable ⇒ continuous, but continuous doesn't imply differentiable.

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12 · Topic 3.6 · When Derivatives Don't Exist

Four ways f'(a) can fail to exist

Failure mode	What you see	Example
Corner / cusp	Left derivative ≠ right derivative	f(x) = \|x\| at x = 0
Vertical tangent	Tangent is vertical → slope undefined	f(x) = ∛x at x = 0
Discontinuity	f(a) undefined or jumps	f(x) = 1/x at x = 0
Oscillating tangent	Near a the curve wiggles without settling	f(x) = x sin(1/x) near 0 (extended)

Don't confuse these with "f(a) is not defined"

If f(a) doesn't exist, then a fortiori f'(a) can't exist. But continuity at a is necessary — not sufficient.

Original comic · ap-study.com

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13 · Topic 3.3 · Power Rule

Power rule · FUN-3.A.1

For any real n ≠ 0, defined on the appropriate domain

d/dx [xⁿ] = n x^{n − 1}

The single most useful formula in calculus. Multiply by the exponent, then drop it by one.

Example 1

d/dx [x⁵] = 5x⁴

Example 2

d/dx [x] = 1 · x⁰ = 1

Example 3

d/dx [x⁻³] = −3x⁻⁴

Example 4

d/dx [√x] = d/dx [x^1/2] = (1/2) x^−1/2

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14 · Topic 3.3 · Constant · Sum · Difference

Three rules that go with power rule

Constant rule

d/dx [c] = 0 for any constant c

Constant multiple

d/dx [c · f(x)] = c · f'(x)

Sum rule

d/dx [f(x) + g(x)] = f'(x) + g'(x)

Difference rule

d/dx [f(x) − g(x)] = f'(x) − g'(x)

Differentiate a polynomial

Q. f(x) = 4x³ − 6x² + 9x − 7. f'(x) = ?

→ f'(x) = 12x² − 12x + 9. Each term uses the constant-multiple + power rule.

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15 · Topic 3.3 · e^x and ln x Derivatives

The two magic exponentials

Why e ≈ 2.71828 is special

d/dx [e^x] = e^x d/dx [ln x] = 1 / x

The exponential's derivative is itself. The log's derivative is its own inverse.

^x 1 tangent at (0, 1) has slope = 1

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16 · Topic 3.4 · sin & cos Derivatives

sin and cos — derived from a unit circle

FUN-3.B · memorize these

d/dx [sin x] = cos x d/dx [cos x] = −sin x

Both results come from the unit circle & the sin/x → 1 limit from Unit 1.

😤

Meme break

Why did cos get kicked out of the library? It couldn't stop being derivative of sin. Everywhere sin goes, cos shows up one quarter-cycle later. Obsessed behaviour.

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17 · Topic 3.4 · tan, sec, csc, cot

Four more from quotient + chain rules

Function	Derivative	Note
tan x = sin x / cos x	sec²x	always ≥ 0
cot x = cos x / sin x	−csc²x	always ≤ 0
sec x = 1 / cos x	sec x · tan x	undefined at x = π/2 + kπ
csc x = 1 / sin x	−csc x · cot x	undefined at x = kπ

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Memory tip: derivatives of "co" functions are negative. tan and sec are paired (one derivative has the other squared/multiplied respectively).

Proof sketch: Each of these follows from the quotient rule + the sin/cos derivative. Try sec yourself: (1/cos x)' = − (1/cos²x) · (−sin x) = sin x / cos²x = sec x · tan x. ✓

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18 · Topic 3.3 · Combining Rules

Combining all the basic rules

Example A — mixed

Q. f(x) = 3x⁴ − 5√x + 4/x + 2e^x + 7 ln x − π ⇒ f'(x) = ?

Step Term by term: 12x³ − (5/2)x^−1/2 − 4x⁻² + 2e^x + 7/x + 0.

Example B — trig + polynomial

Q. g(x) = x³ · sin x. g'(x) = ?

Step This requires the product rule, next slide.

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19 · Power Rule Edge Cases

Power rule pitfalls

Pitfall	Correct application
d/dx [x²/x³]	Simplify first: x⁻¹ → −x⁻²
d/dx [3^x] (NOT 3x^…)	Different rule (e ^x-form): 3^x ln 3. Power rule applies only to xⁿ.
d/dx [e³]	e³ is a constant ≈ 20.09. Derivative = 0, not e³.
d/dx [x^x]	No power-rule shortcut; use logarithmic differentiation (BC).

Watch out for the variable on the bottom

The power rule only applies when the variable is in the base. If the variable appears in the exponent, you need a different formula. (We'll see those in Unit 4 — related rates + exponential/log applications.)

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20 · Topic 3.5 · Product Rule

Product rule · FUN-3.B.4

If u and v are functions of x

d/dx [u · v] = u' · v + u · v'

"Derivative of the first times the second, plus the first times the derivative of the second."

Memory acronym · FOIL

(fg)' = (first)' × (second) + (first) × (second)'

💘

Meme break · pick-up line

"Are you the product rule? Because every time I work with you, I have to take both parts into account — and keep track of who changed."

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21 · Product Rule · Worked

Product rule in action

Three practice runs

Q1. f(x) = x² · sin x. f'(x) = ?

→ 2x · sin x + x² · cos x.

Q2. f(x) = (3x + 1) · e^x. f'(x) = ?

→ 3 · e^x + (3x + 1) · e^x = e^x(3 + 3x + 1) = e^x(3x + 4).

Q3. f(x) = x · ln x. f'(x) = ?

→ 1 · ln x + x · (1/x) = ln x + 1.

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22 · Topic 3.5 · Quotient Rule

Quotient rule · FUN-3.B.5

If u and v are functions of x, with v(x) ≠ 0

d/dx [u / v] = (u' v − u v') / v²

Low-dee-high minus high-dee-low, all over low-squared.

🎵

Memory song: "dee-low-high, minus, hi-dee-low, all over low-low." Or just remember the formula and write it out every time.

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23 · Quotient Rule · Worked

Quotient rule in action

Three practice runs

Q1. f(x) = (3x + 1) / (x² + 5). f'(x) = ?

→ (3(x² + 5) − (3x + 1)(2x)) / (x² + 5)² = (3x² + 15 − 6x² − 2x) / (x² + 5)² = (−3x² − 2x + 15) / (x² + 5)².

Q2. f(x) = sin x / x. f'(x) = ?

→ (x cos x − sin x) / x².

Q3. f(x) = tan x. Verify sec²x using the quotient rule.

→ (cos x · cos x − sin x · (−sin x)) / cos²x = (cos²x + sin²x) / cos²x = 1 / cos²x = sec²x. ✓

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24 · Topic 3.6 · Vertical Tangent & Cusp

Vertical tangent vs cusp

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Difference: Vertical tangent has the curve going up on both sides of one point with infinite slope. A cusp has the curve approaching the point from both sides with opposite vertical directions — like a sharp point.

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25 · Topic 3.7 · Higher Order Derivatives

Higher-order derivatives

Keep differentiating

f(x) → f'(x) → f''(x) → f'''(x) → f⁽ⁿ⁾(x)

Each is the derivative of the previous one. dy/dx, d²y/dx², d³y/dx³.

Example

Q. f(x) = 4x⁴ − 3x² + 5x − 2. Find f'(x), f''(x), f'''(x).

f' 16x³ − 6x + 5

f'' 48x² − 6

f''' 96x

f⁽⁴⁾ 96

f⁽⁵⁾ 0 (we hit the bottom)

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26 · Topic 3.7 · Physical Meaning of Derivatives

Position → Velocity → Acceleration

If s(t) is position (meters, feet, etc.)

v(t) = s'(t) a(t) = v'(t) = s''(t)

First derivative: how fast the position changes. Second derivative: how fast the velocity changes.

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27 · Derivative Rules — Cheat Sheet

Unit 2 cheat sheet — keep visible until slide 38

Function	Derivative	Function	Derivative
c (constant)	0	sin x	cos x
xⁿ	n xⁿ⁻¹	cos x	−sin x
c · f(x)	c · f'(x)	tan x	sec²x
f + g	f' + g'	cot x	−csc²x
e^x	e^x	sec x	sec x · tan x
ln x	1/x	csc x	−csc x · cot x
f · g	f' · g + f · g' (product)	f / g	(f' g − f g') / g² (quotient)

🧠

Need these by reflex. The next six units build everything else on top of these.

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28 · Worked Example 1

Polynomial derivative practice

Differentiate

Q1. f(x) = 6x² + 4x − 9

→ f'(x) = 12x + 4.

Q2. g(x) = 2x³ − 5x²/3 + π (note the π is just a constant)

→ g'(x) = 6x² − (10/3)x + 0 = 6x² − 10x/3.

Q3. h(x) = (x − 1)² + (x + 1)⁻¹

→ h'(x) = 2(x − 1) + (−1)(x + 1)⁻² = 2(x − 1) − 1/(x + 1)².

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29 · Worked Example 2

Product & quotient practice

Use the right rule

Q1. f(x) = x² · e^x. f'(x) = ?

→ Product rule: 2x · e^x + x² · e^x = e^x · (x² + 2x).

Q2. g(x) = (sin x) / (1 + cos x). g'(x) = ?

→ Quotient: (cos x · (1 + cos x) − sin x · (−sin x)) / (1 + cos x)² = (cos x + cos²x + sin²x) / (1 + cos x)² = (cos x + 1) / (1 + cos x)² = 1 / (1 + cos x).

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30 · Worked Example 3 · Higher-Order

Higher-order derivatives

Practice

Q1. f(x) = x³ − 3x² + 5x − 7. Find f'''(x).

→ f'(x) = 3x² − 6x + 5. f''(x) = 6x − 6. f'''(x) = 6.

Q2. g(x) = cos x. Find g⁽⁴⁾(x).

→ g'(x) = −sin x. g''(x) = −cos x. g'''(x) = sin x. g⁽⁴⁾(x) = cos x. (Cycles every 4 derivatives.)

Q3. h(x) = e^x. Find h⁽¹⁰⁰⁾(x).

→ All derivatives are e^x. So h⁽¹⁰⁰⁾(x) = e^x.

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31 · Quick Check

Quick check — derivative rules

d/dx [x · ln x] = ?

A1/x

Bln x

Cln x + 1

Dx · ln x

✅ Answer: C (ln x + 1). Product rule: 1 · ln x + x · (1/x) = ln x + 1.

Click an option to reveal the answer.

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32 · AP-style MCQ

AP MCQ · derivative of a product

If f(x) = x² + 3x and g(x) = sin x, then d/dx [f(x) · g(x)] = ?

A(2x + 3) · cos x

B(x² + 3x) · cos x

C(2x + 3) · sin x + (x² + 3x) · cos x

D(2x + 3) · sin x + (x² + 3x) · (−sin x)

✅ Answer: C. Product rule with f'(x) = 2x + 3 and g'(x) = cos x: f'g + fg'.

Click an option to check.

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33 · AP-style FRQ

FRQ · tangent line

AP-style problem · 4 points

Let f(x) = (x² − 1) · e^x for x ≥ 0.

(a) Find f'(x).

(b) Write the equation of the tangent line to y = f(x) at x = 0.

Solution sketch: (a) f'(x) = 2x · e^x + (x² − 1) · e^x = e^x(x² + 2x − 1). (b) f(0) = −1, f'(0) = −1 → y = −1 − (x − 0) = −x − 1. (c) Set f'(x) = 0 ⇒ x² + 2x − 1 = 0 ⇒ x = −1 ± √2. Only positive: √2 − 1 ≈ 0.414.

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34 · Common Mistakes

Top 5 mistakes students make in Unit 2

① Forgetting the second term in product rule

"(fg)' = f'g" is a one-way ticket to wrong answers. Always include both f'g and fg'.

② Mixing up d/dx [e^x] with d/dx [x^e]

e^x derivative = e^x. x^e derivative = e · x^e−1. The exponent letter vs variable base matters!

③ Saying derivative at a corner "exists"

|x| at x = 0 has left derivative = −1, right derivative = +1. They don't agree → no derivative.

④ Confusing f'(a) with f(a)

f(a) is the y-value. f'(a) is the slope. Very different numbers.

⑤ Treating "1/x" as just "1/x", not "x⁻¹"

It's the latter. d/dx [x⁻¹] = −x⁻² = −1/x².

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35 · Real-World Derivatives

Where the derivative shows up

Velocity (physics): v(t) = s'(t). Speedometer in your car is literally a derivative in real time.
Marginal cost (economics): MC = dC/dQ. The cost of producing "one more unit."
Population growth: dP/dt = rP. The rate a population is growing at any moment — used in epidemiology for R₀ calculations.
Marginal revenue (finance): If revenue is R(q), marginal revenue R'(q) is the change in revenue per additional unit sold.
Pharmacokinetics: dC/dt = how fast a drug concentration changes in your blood; critical for safe dosing.

🤯

Did you know? The accelerometer in your phone is just a tiny mass on a spring — its displacement is integrated twice to get position, derived twice to get jerk. Modern cars use derivatives to detect skidding and adjust brakes in real time.

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36 · YouTube Lessons

Best YouTube for this unit

Three picks — concept, examples, visual.

Concept Derivatives — The Big Picture (3Blue1Brown) 📺 3Blue1Brown⏱ 15 min Examples Product & Quotient Rule — 10 Examples 📺 The Organic Chemistry Tutor⏱ 22 min Visual Why d/dx [eˣ] = eˣ 📺 3Blue1Brown⏱ 13 min

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37 · Unit 2 Recap

Unit 2 in 60 seconds

Derivative = slope of tangent = limit of difference quotient.
Power rule + sum + constant multiple handle all polynomials.
e^x and ln x are the natural exponential pair; their derivatives are themselves and 1/x.
sin and cos come from the unit circle; + their cofunctions.
Product rule (u·v)' = u'v + uv'; Quotient rule (u/v)' = (u'v − uv') / v².
Differentiability ⇒ continuity (but not vice versa).
Higher-order derivatives track position → velocity → acceleration → jerk.

🔗

Next: Open Unit 3 · Composite, Implicit & Inverse Functions from the unit index. The chain rule lets you differentiate any composition — which is what makes derivatives useful across all of physics, economics, and biology.

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38 · Knowledge Coverage Checklist

Unit 2 · everything covered ✓

Mapped 1:1 against the 2024-25 College Board CED for Calculus AB Topic 3.

3.1 / CHA-2.B Defining average rate of change.
3.1 / CHA-2.C Defining instantaneous rate (derivative at a point).
3.1 / CHA-2.C Estimating derivatives from a graph (secant → tangent).
3.1 / CHA-2.C Using the limit definition of the derivative.
3.2 / CHA-2.D Derivative as a function f'(x).
3.3 / FUN-3.A Power rule, constant rule, sum/difference.
3.3 / FUN-3.A Derivatives of e^x and ln x.
3.4 / FUN-3.B Derivatives of sin x, cos x.
3.4 / FUN-3.B Derivatives of tan, cot, sec, csc.
3.5 / FUN-3.B Product rule.
3.5 / FUN-3.B Quotient rule.
3.6 / FUN-3.B Differentiability ⇒ continuity; where derivatives don't exist.
3.7 / CHA-2.D Higher-order derivatives & physical meaning.
Worked examples · 4 unlimited + 3 quiz · 5 practice problems embedded.
Common errors · 5 pitfalls explicitly called out.
3 curated YouTube lessons (concept / example / visual).
Real-world connections · 5 applications across physics, economics, finance, biology.

🎓

Done with Unit 2! Move to Unit 3 when ready — chain rule, implicit differentiation, and inverse derivatives.

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Unit 2 1 / 38 Differentiation · Definition & Basic Rules

Differentiation · Definition& Basic Rules

By the end of this unit, you will be able to…

Unit 2 → AP Calculus AB Topic 3

Key Terms · Part 1

Average rate of change

Instantaneous rate

Derivative f'(a)

Difference quotient

Differentiable at a

Tangent line

Five ways to write the derivative

What does "instantaneous rate" actually look like?

The derivative, formally

Find f'(x) from first principles

Find f'(x) — square root version

The tangent line at a point

Differentiable ⇒ Continuous (not the reverse)

Four ways f'(a) can fail to exist

Don't confuse these with "f(a) is not defined"

Power rule · FUN-3.A.1

Example 1

Example 2

Example 3

Example 4

Three rules that go with power rule

Constant rule

Constant multiple

Sum rule

Difference rule

The two magic exponentials

sin and cos — derived from a unit circle

Four more from quotient + chain rules

Combining all the basic rules

Power rule pitfalls

Watch out for the variable on the bottom

Product rule · FUN-3.B.4

Product rule in action

Quotient rule · FUN-3.B.5

Quotient rule in action

Vertical tangent vs cusp

Higher-order derivatives

Position → Velocity → Acceleration

Unit 2 cheat sheet — keep visible until slide 38

Polynomial derivative practice

Product & quotient practice

Higher-order derivatives

Quick check — derivative rules

AP MCQ · derivative of a product

FRQ · tangent line

Top 5 mistakes students make in Unit 2

① Forgetting the second term in product rule

② Mixing up d/dx [ex] with d/dx [xe]

③ Saying derivative at a corner "exists"

④ Confusing f'(a) with f(a)

⑤ Treating "1/x" as just "1/x", not "x−1"

Where the derivative shows up

Best YouTube for this unit

Unit 2 in 60 seconds

Unit 2 · everything covered ✓

Differentiation · Definition
& Basic Rules

② Mixing up d/dx [e^x] with d/dx [x^e]

⑤ Treating "1/x" as just "1/x", not "x⁻¹"