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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

1. Average Rate of Change

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

$\frac{f(b) - f(a)}{b - a}$

2. Instantaneous Rate of Change

As $b \to a$ , the secant line becomes the tangent line, and we get the derivative:

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

This is the limit definition of the derivative — the most fundamental formula in calculus.

3. Computing Derivatives from the Definition

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

$f'(x) = \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h} = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h} = \lim_{h \to 0} \frac{2xh + h^2}{h} = \lim_{h \to 0} (2x + h) = 2x$

4. Alternate Form

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

This form is useful when given a specific point rather than a general formula.

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Two Forms of the Derivative Definition

Form	Formula	Use When
Standard	$\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$

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Identify the Derivative 🔍

Match each limit to the derivative it represents.

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

∫ Differentiability

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

1. Differentiability Requires Continuity

If $f$ is differentiable at $x = c$ , then $f$ is continuous at $x = c$ .

Part 3: Reading Derivatives from Graphs

∫ Graphical Interpretation of Derivatives

Part 3 of 7 — Reading Derivatives from Graphs

1. Derivative = Slope of Tangent Line

At any point, $f'(a)$ equals the slope of the tangent line to $f$ at $x = a$ .

Part 4: The Language of Derivatives

∫ Derivative Notation

Part 4 of 7 — The Language of Derivatives

1. Common Notations

All of these mean "the derivative of $y$ with respect to $x$ ":

Notation	Read as	Emphasized by
$f'(x)$

Part 5: The Tangent Line Equation

∫ Tangent Lines and Linear Approximation

Part 5 of 7 — The Tangent Line Equation

1. Equation of the Tangent Line

The tangent line to $f$ at $x = a$ has:

Slope: $m = f'(a)$

Part 6: Derivative Definition Practice

∫ Problem-Solving Workshop

Part 6 of 7 — Derivative Definition Practice

Strategy: Limit-Definition Problems

When asked to find a derivative using the limit definition:

Write out $\frac{f(x+h)-f(x)}{h}$
Expand carefully

Part 7: Comprehensive Review

∫ Review & Applications

Part 7 of 7 — Comprehensive Review

The Big Picture

The derivative $f'(a)$ answers: "How fast is $f$ changing at $x = a$ ?"

Geometrically: slope of the tangent line

limh→0h(x+h)2−x2=

limh→0hx2+2xh+h2−x2=

AP Exam note: You may be given a limit and asked to identify it as a derivative. For example, $\lim_{h \to 0} \frac{\sin(\pi/6 + h) - 1/2}{h}$ represents $f'(\pi/6)$ where $f(x) = \sin x$ .

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

⚠️ The converse is false: $f(x) = |x|$ is continuous at $x = 0$ but NOT differentiable there.

2. When Derivatives Fail to Exist

The derivative does not exist at:

Corners/cusps: $f(x) = |x|$ at $x = 0$ (left slope $= -1$ , right slope $= 1$ )
Vertical tangent lines: $f(x) = x^{1/3}$ at $x = 0$ (slope $\to \pm\infty$ )
Discontinuities: Any type of discontinuity
Endpoints: Only one-sided derivative exists

3. Checking Differentiability for Piecewise Functions

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

Check continuity: $\lim_{x \to 1^-} x^2 = 1 = \lim_{x \to 1^+} (2x-1) = 1$ ✓
Check derivatives match: Left: $f'(x) = 2x \to 2(1) = 2$ . Right: $f'(x) = 2$ . ✓

Both derivatives equal 2, so $f$ IS differentiable at $x = 1$ .

A differentiable function "looks like a line" when you zoom in enough. This is the geometric meaning of differentiability — the graph has no sharp turns or breaks at the microscopic level.

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Quick Reference: Differentiability vs. Continuity

Function	Continuous at $x=0$ ?	Differentiable at $x=0$ ?
$f(x) = x^2$	Yes	Yes
$f(x) = \\|x\\|$	Yes	No (corner)
$f(x) = x^{1/3}$	Yes	No (vertical tangent)
$f(x) = 1/x$	No	No

Memory aid: Differentiable ⟹ Continuous, but Continuous ⟹ Differentiable is FALSE.

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Differentiability Check 🔍

f'(a) > 0

f

is increasing at

a

f'(a) < 0

f

is decreasing at

a

f'(a) = 0

f

has a horizontal tangent at

a

(possible max, min, or inflection)

2. From Graph of $f$ to Graph of $f'$

Feature of $f$	Corresponding feature of $f'$
$f$ increasing	$f' > 0$ (above $x$ -axis)
$f$ decreasing	$f' < 0$ (below $x$ -axis)
Local max of $f$	$f' = 0$ (crosses from $+$ to $-$ )
Local min of $f$	$f' = 0$ (crosses from $-$ to $+$ )
Inflection point of $f$	Local max or min of $f'$
$f$ concave up	$f'$ increasing
$f$ concave down	$f'$ decreasing

3. Estimating Derivatives from Data

From a table of values, approximate $f'(a)$ using the symmetric difference quotient:

$f'(a) \approx \frac{f(a+h) - f(a-h)}{2h}$

This is more accurate than the one-sided difference quotient.

4. Reading $f$ from $f'$

Given the graph of $f'$ :

Where $f' > 0$ , $f$ is increasing
Where $f' < 0$ , $f$ is decreasing
Where $f'$ changes sign, $f$ has a local extremum

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AP Exam Graph-Reading Tips

When given the graph of $f'$ and asked about $f$ :

Zeros of $f'$ = horizontal tangent lines of $f$ (possible extrema)
Sign changes of $f'$ = extrema of $f$
Extrema of $f'$ = inflection points of $f$
$f'$ positive = $f$ rising, $f'$ negative = $f$ falling

Common trap: A zero of $f'$ is NOT always an extremum. If $f'$ doesn't change sign (like $f(x) = x^3$ at ), it's just an inflection point.

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From $f$ to $f'$ 🔍

Determine the sign of $f'$ at each point.

2. Leibniz Notation: More Than Just a Symbol

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

Chain rule: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$ (cancels like fractions!)

Evaluated at a point: $\left.\frac{dy}{dx}\right|_{x=3}$ means "evaluate the derivative at $x = 3$ "

3. Higher-Order Derivatives

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

4. Units of Derivatives

If $y$ has units of meters and $x$ has units of seconds, then:

$\frac{dy}{dx}$ has units of $\frac{\text{meters}}{\text{seconds}}$ (velocity)
$\frac{d^2y}{dx^2}$ has units of $\frac{\text{meters}}{\text{seconds}^2}$ (acceleration)

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When to Use Which Notation

$f'(x)$ — Best for general function manipulation, stating rules
$\frac{dy}{dx}$ — Best for related rates, implicit differentiation, chain rule
$\frac{d}{dx}[\text{expression}]$ — Best as an operator: "take the derivative of this expression"

Example in context: "Find $\frac{d}{dx}[x^2 \sin x]$ " means "differentiate $x^2 \sin x$ with respect to ."

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Match the Notation 🔍

Point:

(a, f(a))

Point-slope form: $y - f(a) = f'(a)(x - a)$

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

$f(2) = 8$ , so the point is $(2, 8)$
$f'(x) = 3x^2$ , so $f'(2) = 12$
Tangent line: $y - 8 = 12(x - 2)$ → $y = 12x - 16$

The normal line is perpendicular to the tangent line. If the tangent slope is $m$ , the normal slope is $-\frac{1}{m}$ (negative reciprocal).

4. Tangent Line as a Local Approximation

Near $x = a$ , the function $f(x)$ is well-approximated by its tangent line:

$f(x) \approx f(a) + f'(a)(x - a)$

This is called linearization or linear approximation.

Example: Approximate $\sqrt{4.1}$ using the tangent line to $f(x) = \sqrt{x}$ at $x = 4$ :

$f(4) = 2$ , $f'(x) = \frac{1}{2\sqrt{x}}$ , $f'(4) = \frac{1}{4}$

$\sqrt{4.1} \approx 2 + \frac{1}{4}(4.1 - 4) = 2 + 0.025 = 2.025$

(Actual: $\sqrt{4.1} = 2.02485...$ )

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Tangent Line Checklist

Find $f(a)$ — the $y$ -value at the point
Find $f'(a)$ — the slope at the point
Write: $y - f(a) = f'(a)(x - a)$

Is the Approximation an Over- or Under-estimate?

If $f$ is concave up near $a$ : tangent line is below the curve → underestimate
If $f$ is concave down near $a$ : tangent line is above the curve → overestimate

This is a common AP FRQ follow-up question!

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Tangent Line Practice 🔍

Simplify — everything should cancel the

h

in the denominator

Take

\lim_{h \to 0}

Worked Example: $f(x) = \frac{1}{x}$

$f'(x) = \lim_{h \to 0} \frac{\frac{1}{x+h} - \frac{1}{x}}{h} = \lim_{h \to 0} \frac{\frac{x-(x+h)}{x(x+h)}}{h} = \lim_{h \to 0} \frac{-h}{hx(x+h)} = \lim_{h \to 0} \frac{-1}{x(x+h)} = \frac{-1}{x^2}$

Worked Example: Recognizing Derivative Limits

"Find $\lim_{h \to 0} \frac{\cos(\pi + h) - \cos(\pi)}{h}$ "

This IS $f'(\pi)$ where $f(x) = \cos x$ . So the answer is $f'(\pi) = -\sin(\pi) = 0$ .

Much faster than trying to expand $\cos(\pi + h)$ !

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Recognizing Derivatives in Disguise

Limit Expression	Recognized As	Answer
$\lim_{h \to 0} \frac{e^{2+h}-e^2}{h}$	$f'(2)$ , $f(x)=e^x$	$e^2$
$\lim_{x \to 3} \frac{x^2-9}{x-3}$
$\lim_{h \to 0} \frac{\ln(1+h)}{h}$	, ? No: ,

These "recognize the derivative" problems save huge amounts of computation on the AP exam.

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Recognize & Evaluate 🔍

Physically: instantaneous rate of change

Algebraically:

\lim_{h \to 0} \frac{f(a+h)-f(a)}{h}

$\text{Position } s(t) \xrightarrow{\text{derivative}} \text{Velocity } v(t) = s'(t) \xrightarrow{\text{derivative}} \text{Acceleration } a(t) = v'(t) = s''(t)$

Differentiability Hierarchy

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

But NONE of the reverse implications hold!

Derivative definition: $f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$
Alternate form: $f'(a) = \lim_{x \to a} \frac{f(x)-f(a)}{x-a}$
Tangent line: $y = f(a) + f'(a)(x-a)$
Linear approximation: $f(x) \approx f(a) + f'(a)(x-a)$ for near

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Common AP Exam Derivative Questions

"Find $f'(a)$ using the definition" — Must show the limit, not just use shortcut rules
"What does $f'(3) = -2$ mean in context?" — At $x = 3$ , $f$ is decreasing at a rate of 2 [units] per [unit]
"Is $f$ differentiable at $x = c$ ?" — Check continuity AND matching derivatives from both sides
"Find the tangent/normal line" — Use point-slope form with $f(a)$ and $f'(a)$
"Approximate using linearization" — $f(x) \approx f(a) + f'(a)(x-a)$

Check Your Understanding 🎯

Final Review 🔍

Definition of the Derivative

2. When Derivatives Fail to Exist

3. Checking Differentiability for Piecewise Functions

4. Local Linearity

Quick Reference: Differentiability vs. Continuity

2. From Graph of $f$ to Graph of $f'$

3. Estimating Derivatives from Data

4. Reading $f$ from $f'$

AP Exam Graph-Reading Tips

2. Leibniz Notation: More Than Just a Symbol

3. Higher-Order Derivatives

4. Units of Derivatives

When to Use Which Notation

2. Worked Example

3. Normal Line

4. Tangent Line as a Local Approximation

Tangent Line Checklist

Is the Approximation an Over- or Under-estimate?

Worked Example: $f(x) = \frac{1}{x}$

Worked Example: Recognizing Derivative Limits

Recognizing Derivatives in Disguise

Key Relationships

Differentiability Hierarchy

Essential Formulas

Common AP Exam Derivative Questions

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

1. Average Rate of Change

2. Instantaneous Rate of Change

3. Computing Derivatives from the Definition

4. Alternate Form

Two Forms of the Derivative Definition

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

∫ Differentiability

1. Differentiability Requires Continuity

Part 3: Reading Derivatives from Graphs

∫ Graphical Interpretation of Derivatives

1. Derivative = Slope of Tangent Line

Part 4: The Language of Derivatives

∫ Derivative Notation

1. Common Notations

Part 5: The Tangent Line Equation

∫ Tangent Lines and Linear Approximation

1. Equation of the Tangent Line

Part 6: Derivative Definition Practice

∫ Problem-Solving Workshop

Strategy: Limit-Definition Problems

Part 7: Comprehensive Review

∫ Review & Applications

The Big Picture

2. When Derivatives Fail to Exist

3. Checking Differentiability for Piecewise Functions

4. Local Linearity

Quick Reference: Differentiability vs. Continuity

2. From Graph of fff to Graph of f′f'f′

3. Estimating Derivatives from Data

4. Reading fff from f′f'f′

AP Exam Graph-Reading Tips

2. Leibniz Notation: More Than Just a Symbol

3. Higher-Order Derivatives

4. Units of Derivatives

When to Use Which Notation

2. Worked Example

3. Normal Line

4. Tangent Line as a Local Approximation

Tangent Line Checklist

Is the Approximation an Over- or Under-estimate?

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

Worked Example: Recognizing Derivative Limits

Recognizing Derivatives in Disguise

Key Relationships

Differentiability Hierarchy

Essential Formulas

Common AP Exam Derivative Questions

2. From Graph of $f$ to Graph of $f'$

4. Reading $f$ from $f'$

Worked Example: $f(x) = \frac{1}{x}$