🎯⭐ INTERACTIVE LESSON

Definition of the Derivative

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Definition of the Derivative - Complete Interactive Lesson

Part 1: From Average to Instantaneous Rate of Change

∫ The Derivative as a Limit

Part 1 of 7 — From Average to Instantaneous Rate of Change

Topics in This Part

Section
📖 Average Rate of Change (Secant Lines)
Instantaneous Rate of Change (Tangent Lines)
📌 The Limit Definition of the Derivative
Computing Derivatives from the Definition
Alternate Form of the Definition

🔑 Key Concept: The derivative $f'(a)$ is the instantaneous rate of change of $f$ at $x = a$ , defined as the limit of average rates of change as the interval shrinks to zero.

📖 Average Rate of Change

The average rate of change of $f$ on $[a,b]$ is the slope of the secant line through $(a, f(a))$ and $(b, f(b))$ :

📌 The Limit Definition of the Derivative

As $h \to 0$ , the secant line becomes the tangent line:

$\boxed{f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}}$

Check Your Understanding 🎯

Alternate Form of the Derivative

$\boxed{f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}}$

Recognizing Derivatives 🎯

Identify the Derivative 🔍

Match each limit to the derivative it represents.

Compute from the Definition ✍️

Part 2: When Derivatives Exist (and When They Don't)

∫ Differentiability

Part 2 of 7 — When Derivatives Exist (and When They Don't)

Topics in This Part

Section
📖 Differentiability Implies Continuity
Four Ways Derivatives Fail to Exist
📌 Piecewise Differentiability Check
Local Linearity

🔑 Key Concept: Differentiability is STRONGER than continuity. Every differentiable function is continuous, but not every continuous function is differentiable.

📖 Differentiability ⟹ Continuity

$\boxed{f \text{ differentiable at } c \implies f \text{ continuous at } c}$

Part 3: Reading Derivatives from Graphs

∫ Graphical Interpretation of Derivatives

Part 3 of 7 — Reading Derivatives from Graphs

Topics in This Part

Section
📖 Derivative = Slope of Tangent Line
From Graph of $f$ to Graph of $f'$
📌 Estimating Derivatives from Tables
Reading $f$ from (Reverse Direction)

Part 4: The Language of Derivatives

∫ Derivative Notation & Units

Part 4 of 7 — The Language of Derivatives

Topics in This Part

Section
📖 The Four Common Notations
Leibniz Notation — Why It's Special
📌 Higher-Order Derivatives
Units of Derivatives
Interpreting Derivatives in Context

🔑 Key Concept: Different notations emphasize different aspects of the derivative. Leibniz notation ( $dy/dx$ ) is especially powerful because it suggests the chain rule and carries units naturally.

📖 The Four Common Notations

Notation	Name	Read As	Best For

Part 5: The Tangent Line Equation

∫ Tangent Lines and Linear Approximation

Part 5 of 7 — The Tangent Line Equation

Topics in This Part

Section
📖 Equation of the Tangent Line
Normal Lines (Perpendicular)
📌 Linear Approximation (Linearization)
Over- vs. Under-Estimates

🔑 Key Concept: The tangent line at $x = a$ is the best linear approximation to $f$ near $a$ . This idea is the foundation of differential calculus.

Part 6: Derivative Definition Practice

∫ Problem-Solving Workshop

Part 6 of 7 — Derivative Definition Practice

Strategy Guide

Problem Type	Method
"Find $f'(x)$ using the definition"	Write the limit, expand, simplify, cancel $h$ , evaluate
"Evaluate this limit" (looks like a derivative)	Recognize as $f^{'}$ , use rules instead

Part 7: Comprehensive Review

∫ Review & AP Exam Applications

Part 7 of 7 — Comprehensive Review

The Derivative: Three Perspectives

Perspective	Interpretation
Geometric	Slope of the tangent line to $f$ at $x = a$
Physical	Instantaneous rate of change of $f$ at

$\boxed{\text{AROC} = \frac{f(b) - f(a)}{b - a}}$

Physical interpretation: If $f(t)$ = position at time $t$ , then AROC is the average velocity on $[a,b]$ .

Example: Average Velocity

A car's position is $s(t) = t^2$ meters at time $t$ seconds.

Average velocity from $t=1$ to $t=3$ :

$\frac{s(3)-s(1)}{3-1} = \frac{9-1}{2} = 4 \text{ m/s}$

But what is the velocity at exactly $t = 1$ ? We need to let the interval shrink...

🔑 Key Idea: Average rate of change → secant line slope. Make the interval infinitely small → tangent line slope.

This is the most fundamental formula in calculus.

Computing $f'(x)$ from the Definition

Example: Find $f'(x)$ for $f(x) = x^2$ .

Step	Computation
Write the limit	$f'(x) = \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h}$
Expand	$= \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h}$
Cancel	$= \lim_{h \to 0} \frac{2xh + h^2}{h}$
Factor out $h$	$= \lim_{h \to 0} (2x + h)$
Evaluate	$= 2x$

$\boxed{f(x) = x^2 \implies f'(x) = 2x}$

AP Tip: On the AP exam, you MUST show the limit process — you cannot just write down the answer using shortcut rules when asked to use the definition.

Standard Form	Alternate Form
$\lim_{h \to 0} \frac{f(a+h)-f(a)}{h}$	$\lim_{x \to a} \frac{f(x)-f(a)}{x-a}$
Uses increment $h$	Uses the point $x$ directly
Best for: finding $f'(x)$ as a function	Best for: evaluating $f'(a)$ at a specific point

Recognizing Derivatives in Disguise

On the AP exam, you may be given a limit and asked to identify it as a derivative:

Example: $\lim_{h \to 0} \frac{\sin(\pi/6 + h) - 1/2}{h}$

This is $f'(\pi/6)$ where $f(x) = \sin x$ , since $\sin(\pi/6) = 1/2$ .

Answer: $f'(\pi/6) = \cos(\pi/6) = \frac{\sqrt{3}}{2}$

AP Tip: If you see a limit that looks like $\frac{f(a+h)-f(a)}{h}$ or $\frac{f(x)-f(a)}{x-a}$ , identify $f$ and $a$ first — then use derivative rules instead of computing the limit directly.

Contrapositive: If $f$ is NOT continuous at $c$ , then $f$ is NOT differentiable at $c$ .

Warning: The converse is FALSE!

$f(x) = |x|$ is continuous at $x = 0$ but NOT differentiable.

$\text{Differentiable} \implies \text{Continuous} \implies \text{Limit Exists}$

None of these arrows reverse! Each arrow is a one-way implication.

AP Tip: "Differentiable ⟹ Continuous" appears on nearly every AP exam. Know it cold, and remember the converse is false.

Four Ways Derivatives Fail to Exist

Type	What Happens	Example	At
Corner	Left and right slopes differ	$f(x) =	x
Cusp	Slopes → $\pm\infty$ from opposite sides	$f(x) = x^{2/3}$	$x = 0$
Vertical tangent	Slope → $\pm\infty$ from same side	$f(x) = x^{1/3}$	$x = 0$
Discontinuity	Function jumps or is undefined	$f(x) = \lfloor x \rfloor$	$x = n$

Corner: $f(x) = |x|$ at $x = 0$

$f'(0^-) = \lim_{h \to 0^-} \frac{|0+h|-|0|}{h} = \lim_{h \to 0^-} \frac{-h}{h} = -1$

$f'(0^+) = \lim_{h \to 0^+} \frac{|0+h|-|0|}{h} = \lim_{h \to 0^+} \frac{h}{h} = 1$

Since $-1 \neq 1$ , $f'(0)$ does not exist.

🔑 Key Fact: At a corner, the function is continuous but the left and right derivatives are different finite numbers.

Check Your Understanding 🎯

📌 Checking Differentiability for Piecewise Functions

Two-Step Process

Step 1: Check continuity (necessary condition)

Evaluate left and right limits at the breakpoint

Step 2: Check that derivatives match (sufficient condition)

Compute derivatives of each piece and evaluate at the breakpoint

Example: $f(x) = \begin{cases} x^2 & x \leq 1 \\ 2x - 1 & x > 1 \end{cases}$

Check	Left Piece	Right Piece	Match?
Continuity	$\lim_{x \to 1^-} x^2 = 1$

Both pass → $f$ IS differentiable at $x = 1$ .

Example: $g(x) = \begin{cases} x^2 & x \leq 1 \\ 3x - 2 & x > 1 \end{cases}$

Check	Left Piece	Right Piece	Match?
Continuity	$\lim = 1$	$\lim = 1$	✓
Derivative	$2x \to 2$

Continuous but not differentiable at $x = 1$ (corner).

AP Tip: For piecewise functions, ALWAYS check continuity FIRST. If it fails, stop — the function is not differentiable.

Piecewise Differentiability 🎯

Differentiability Check 🔍

Find the Value ✍️

🔑 Key Concept: The derivative gives the slope of the tangent line. Positive derivative means increasing; negative means decreasing; zero means horizontal tangent.

📖 From Graph of $f$ to Graph of $f'$

This is one of the most important skills on the AP exam:

Feature of $f$	Corresponding Feature of $f'$
$f$ increasing	$f' > 0$ (above $x$ -axis)
$f$ decreasing	(below -axis)

Common Trap

$\boxed{f'(c) = 0 \text{ does NOT always mean } f \text{ has an extremum at } c}$

Example: $f(x) = x^3$ at $x = 0$ : $f'(0) = 0$ but doesn't change sign → inflection point, NOT an extremum.

AP Tip: On graph-matching problems, always check: (1) where $f$ has horizontal tangents → $f' = 0$ , (2) where $f$ is steepest → $f'$ peaks, (3) inflection points of → extrema of .

Graph Reading 🎯

📌 Estimating Derivatives from Data Tables

When given a table of values (common on AP FRQ):

Method	Formula	Accuracy
Forward difference	$f'(a) \approx \frac{f(a+h)-f(a)}{h}$	Good
Backward difference	$f'(a) \approx \frac{f(a)-f(a-h)}{h}$
Symmetric (central)	$f'(a) \approx \frac{f(a+h)-f(a-h)}{2h}$

Example with a Table

$x$	0	1	2	3	4
$f(x)$	5	8	13	20	29

Estimate $f'(2)$ :

$f'(2) \approx \frac{f(3)-f(1)}{3-1} = \frac{20-8}{2} = 6$

AP Tip: When estimating derivatives from tables, use the symmetric difference quotient whenever possible. The AP exam scoring guidelines explicitly prefer this method.

The First Derivative Test

$\boxed{\text{Sign change of } f' \text{ at } c \implies \text{local extremum of } f \text{ at } c}$

$f'$ Before $c$	$f'$ After $c$

🔑 Key Fact: The First Derivative Test is the primary tool for classifying critical points on the AP exam.

First Derivative Test 🎯

From $f$ to $f'$ 🔍

Estimate from Data ✍️

Leibniz Notation: More Than a Symbol

$\frac{dy}{dx}$ is NOT a fraction, but it behaves like one in key situations:

$\boxed{\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \quad \text{(Chain Rule — "cancels" like fractions!)}}$

Evaluated at a point: $\left.\frac{dy}{dx}\right|_{x=3}$ means $f'(3)$

AP Tip: The AP exam uses all notations interchangeably. Be comfortable reading $f'(x)$ , $\frac{dy}{dx}$ , and $\frac{d}{dx}[f(x)]$ — they all mean the same thing.

📌 Higher-Order Derivatives

Order	Lagrange	Leibniz	Meaning
First	$f'(x)$	$\frac{dy}{dx}$	Slope / rate of change
Second	$f''(x)$	$\frac{d^2y}{dx^2}$
Third	$f'''(x)$	$\frac{d^3y}{dx^3}$
$n$ -th	$f^{(n)}(x)$	$\frac{d^ny}{dx^n}$

Physical Interpretation Chain

$\text{Position } s(t) \xrightarrow{d/dt} \text{Velocity } v(t) \xrightarrow{d/dt} \text{Acceleration } a(t)$

$s'(t) = v(t), \qquad s''(t) = v'(t) = a(t)$

🔑 Key Fact: The second derivative tells you about concavity: $f'' > 0$ → concave up, $f'' < 0$ → concave down.

Notation & Higher Derivatives 🎯

Units of Derivatives

$\boxed{\text{Units of } \frac{dy}{dx} = \frac{\text{units of } y}{\text{units of } x}}$

Context	$y$ Units	$x$ Units	$dy/dx$ Units	Meaning
Position vs time	meters	seconds	m/s	Velocity
Water volume vs time	gallons	minutes	gal/min	Flow rate
Cost vs quantity

AP Tip: On AP FRQ, you MUST include units when interpreting a derivative in context. "At $t = 5$ hours, the temperature is changing at a rate of $-3$ degrees per hour."

Interpreting Derivatives 🎯

Match the Notation 🔍

Interpret in Context ✍️

📖 Equation of the Tangent Line

$\boxed{y - f(a) = f'(a)(x - a)}$

Three ingredients:

The point: $(a, f(a))$
The slope: $m = f'(a)$
Plug into point-slope form

Worked Example

Find the tangent line to $f(x) = x^3$ at $x = 2$ .

Step	Computation
Point	$f(2) = 8$ → $(2, 8)$
Slope	$f'(x) = 3x^2$ →

Normal Line

The normal line is perpendicular to the tangent. If tangent slope is $m$ :

$\boxed{\text{Normal slope} = -\frac{1}{m}}$

For the example above: normal slope $= -\frac{1}{12}$ , so $y - 8 = -\frac{1}{12}(x-2)$ .

AP Tip: Normal lines appear less frequently than tangent lines, but they do show up! Remember: perpendicular slopes are negative reciprocals.

Tangent Lines 🎯

📌 Linear Approximation (Linearization)

Near $x = a$ , the tangent line approximates the function:

$\boxed{f(x) \approx L(x) = f(a) + f'(a)(x - a)}$

$L(x)$ is called the linearization of $f$ at $x = a$ .

Example: Approximate $\sqrt{4.1}$

Using $f(x) = \sqrt{x}$ at $a = 4$ :

Component	Value
$f(a) = f(4)$	$2$
$f'(x) = \frac{1}{2\sqrt{x}}$

The approximation is excellent for small $\Delta x = x - a$ .

🔑 Key Fact: Linear approximation works best when $x$ is close to $a$ . The farther away, the worse the approximation.

Over- vs. Under-Estimates

$\boxed{\text{Concave up} \implies \text{tangent below curve} \implies \text{underestimate}}$ $\boxed{\text{Concave down} \implies \text{tangent above curve} \implies \text{overestimate}}$

Concavity	$f''$ Sign	Tangent Relative to Curve	Approximation Is
Concave up	$f'' > 0$

Example

For $\sqrt{4.1}$ : $f''(x) = -\frac{1}{4x^{3/2}} < 0$ → concave down → tangent is → overestimate.

Indeed: $2.025 > 2.02485...$ ✓

AP Tip: "Is this an overestimate or underestimate? Justify your answer." is a classic AP FRQ follow-up. Always cite concavity ( $f''$ sign).

Linear Approximation 🎯

Tangent Line Practice 🔍

Linear Approximation ✍️

🔑 Key Principle: Many limit problems are derivatives in disguise. Recognizing this saves massive computation.

📖 Worked Example: $f(x) = \frac{1}{x}$ from the Definition

$f'(x) = \lim_{h \to 0} \frac{\frac{1}{x+h} - \frac{1}{x}}{h}$

Step	Work
Common denominator	$= \lim_{h \to 0} \frac{\frac{x - (x+h)}{x(x+h)}}{h}$

$\boxed{f(x) = \frac{1}{x} \implies f'(x) = -\frac{1}{x^2}}$

Worked Example: $f(x) = \sqrt{x}$ from the Definition

$f'(x) = \lim_{h \to 0} \frac{\sqrt{x+h} - \sqrt{x}}{h} \cdot \frac{\sqrt{x+h}+\sqrt{x}}{\sqrt{x+h}+\sqrt{x}}$

$= \lim_{h \to 0} \frac{(x+h)-x}{h(\sqrt{x+h}+\sqrt{x})} = \lim_{h \to 0} \frac{1}{\sqrt{x+h}+\sqrt{x}} = \frac{1}{2\sqrt{x}}$

AP Tip: For radicals, always conjugate-multiply. For fractions, find common denominators. These are the two key algebraic moves.

Definition Practice 🎯

📌 Recognizing Derivatives in Disguise

$\boxed{\text{If a limit looks like } \frac{f(a+h)-f(a)}{h} \text{ or } \frac{f(x)-f(a)}{x-a}, \text{ identify } f \text{ and } a.}$

Limit Expression	$f(x)$	$a$	$f'(a)$	Answer

🔑 Key Fact: This technique turns hard limit computations into easy derivative evaluations. Look for this pattern on EVERY limit problem!

Recognize & Evaluate 🎯

Mixed Practice 🔍

Recognize the Derivative ✍️

🔑 Key Principle: Mastering the definition of the derivative means understanding ALL three perspectives and knowing when each is most useful.

📖 Complete Formulas Reference

$\boxed{f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} = \lim_{x \to a} \frac{f(x)-f(a)}{x-a}}$

$\boxed{\text{Tangent: } y - f(a) = f'(a)(x-a)}$

$\boxed{\text{Linear approx: } f(x) \approx f(a) + f'(a)(x-a)}$

The Differentiability Hierarchy

$\text{Differentiable} \implies \text{Continuous} \implies \text{Limit Exists}$

None reverse! $|x|$ is continuous but not differentiable. $\lfloor x \rfloor$ has limits from one side but isn't continuous.

Motion Connections

$s(t) \xrightarrow{d/dt} v(t) = s'(t) \xrightarrow{d/dt} a(t) = s''(t)$

Concept	Meaning
$v(t) = 0$	Particle at rest
$v(t) > 0$	Moving right/up
$v(t) < 0$

Comprehensive Review 🎯

📌 Common AP Exam Question Types

Question Type	What to Do
"Find $f'(a)$ using the definition"	Write the limit, expand, simplify, cancel $h$ , evaluate
" $f'(3) = -2$ . Interpret in context."	"At $x=3$ , $f$ is decreasing at 2 [units] per [unit]"
"Is $f$ differentiable at $c$ ?"	Check continuity AND left/right derivatives
"Find the tangent line"	$y - f(a) = f'(a)(x-a)$
"Approximate $f(x)$ "	Linear approx: $f(a) + f'(a)(x-a)$
"Over or underestimate?"	Check $f''$ sign (concavity)
"Evaluate this limit"	Is it a derivative in disguise? Identify $f$ and $a$

AP FRQ Interpretation Template

"At time $t = [\text{value}]$ [units], the [quantity] is [increasing/decreasing] at a rate of $|f'(a)|$ [units of $y$ ] per [units of ]."

Example: If $T(t)$ = temperature (°C) at time $t$ (hours) and $T'(3) = -1.5$ :

"At $t = 3$ hours, the temperature is decreasing at a rate of 1.5 degrees Celsius per hour."

AP Tip: You MUST include units, state increasing/decreasing, and use "rate of" language. This is worth 1–2 points on every contextual interpretation question.

AP Exam Practice 🎯

Final Review 🔍

Tangent Line Application ✍️

h∣0+h∣−∣0∣

\lim_{h \to 0} \frac{e^{2+h}-e^2}{h}

lim_{h \to 0} \frac{e ^{2 + h} - e ^{2}}{h}

$+$	$-$	Local maximum at $c$
$-$	$+$	Local minimum at $c$
$+$	$+$	No extremum (increasing through $c$ )
$-$	$-$	No extremum (decreasing through $c$ )

Definition of the Derivative

Definition of the Derivative - Complete Interactive Lesson

Part 1: From Average to Instantaneous Rate of Change

∫ The Derivative as a Limit

Topics in This Part

📖 Average Rate of Change

📌 The Limit Definition of the Derivative

Alternate Form of the Derivative

Part 2: When Derivatives Exist (and When They Don't)

∫ Differentiability

Topics in This Part

📖 Differentiability ⟹ Continuity

Part 3: Reading Derivatives from Graphs

∫ Graphical Interpretation of Derivatives

Topics in This Part

Part 4: The Language of Derivatives

∫ Derivative Notation & Units

Topics in This Part

📖 The Four Common Notations

Part 5: The Tangent Line Equation

∫ Tangent Lines and Linear Approximation

Topics in This Part

Part 6: Derivative Definition Practice

∫ Problem-Solving Workshop

Strategy Guide

Part 7: Comprehensive Review

∫ Review & AP Exam Applications

The Derivative: Three Perspectives

Example: Average Velocity

Computing f′(x)f'(x)f′(x) from the Definition

Recognizing Derivatives in Disguise

The Hierarchy

Four Ways Derivatives Fail to Exist

Corner: f(x)=∣x∣f(x) = |x|f(x)=∣x∣ at x=0x = 0x=0

📌 Checking Differentiability for Piecewise Functions

Two-Step Process

Example: f(x)={x2x≤12x−1x>1f(x) = \begin{cases} x^2 & x \leq 1 \\ 2x - 1 & x > 1 \end{cases}f(x)={x22x−1​x≤1x>1​

Example: g(x)={x2x≤13x−2x>1g(x) = \begin{cases} x^2 & x \leq 1 \\ 3x - 2 & x > 1 \end{cases}g(x)={x23x−

📖 From Graph of fff to Graph of f′f'f′

Common Trap

📌 Estimating Derivatives from Data Tables

Example with a Table

The First Derivative Test

Leibniz Notation: More Than a Symbol

📌 Higher-Order Derivatives

Physical Interpretation Chain

Units of Derivatives

📖 Equation of the Tangent Line

Worked Example

Normal Line

📌 Linear Approximation (Linearization)

Example: Approximate 4.1\sqrt{4.1}4.1​

Over- vs. Under-Estimates

Example

📖 Worked Example: f(x)=1xf(x) = \frac{1}{x}f(x)=x1​ from the Definition

Worked Example: f(x)=xf(x) = \sqrt{x}f(x)=x​ from the Definition

📌 Recognizing Derivatives in Disguise

📖 Complete Formulas Reference

The Differentiability Hierarchy

Motion Connections

📌 Common AP Exam Question Types

AP FRQ Interpretation Template

Computing $f'(x)$ from the Definition

Corner: $f(x) = |x|$ at $x = 0$

Example: $f(x) = \begin{cases} x^2 & x \leq 1 \\ 2x - 1 & x > 1 \end{cases}$

Example: $g(x) = \begin{cases} x^2 & x \leq 1 \\ 3x - 2 & x > 1 \end{cases}$

📖 From Graph of $f$ to Graph of $f'$

Example: Approximate $\sqrt{4.1}$

📖 Worked Example: $f(x) = \frac{1}{x}$ from the Definition

Worked Example: $f(x) = \sqrt{x}$ from the Definition