Definition of the Derivative

∫ The Derivative as a Limit

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

Topics in This Part

🔑 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 $(b, f(b))$:

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

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.

📌 The Limit Definition of the Derivative

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

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

This is the most fundamental formula in calculus.

Computing $f'(x)$ from the Definition

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

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

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.

Check Your Understanding 🎯

Alternate Form of the Derivative

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

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.

Recognizing Derivatives 🎯

Identify the Derivative 🔍

Match each limit to the derivative it represents.

Compute from the Definition ✍️

∫ Differentiability

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

Topics in This Part

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

📖 Differentiability ⟹ Continuity

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

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.

The Hierarchy

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

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

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

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

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

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 ✍️

∫ Graphical Interpretation of Derivatives

Part 3 of 7 — Reading Derivatives from Graphs

Topics in This Part

🔑 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:

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 $f'$ 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 $f$ → extrema of $f'$.

Graph Reading 🎯

📌 Estimating Derivatives from Data Tables

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

Example with a Table

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

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

🔑 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 ✍️

∫ Derivative Notation & Units

Part 4 of 7 — The Language of Derivatives

Topics in This Part

🔑 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

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

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

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

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 ✍️

∫ Tangent Lines and Linear Approximation

Part 5 of 7 — The Tangent Line Equation

Topics in This Part

🔑 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.

📖 Equation of the Tangent Line

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

Three ingredients:

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

Worked Example

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

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 $a = 4$:

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

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

Example

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

∫ Problem-Solving Workshop

Part 6 of 7 — Derivative Definition Practice

Strategy Guide

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

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

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

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

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

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

🔑 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 ✍️

∫ Review & AP Exam Applications

Part 7 of 7 — Comprehensive Review

The Derivative: Three Perspectives

🔑 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

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

Comprehensive Review 🎯

📌 Common AP Exam Question Types

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 $x$]."

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 ✍️