∫ 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

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 $f(x) = \sin x$.

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Identify the Derivative 🔍

Match each limit to the derivative it represents.

∫ 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 $x = c$.

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

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

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

4. Local Linearity

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

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

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Differentiability Check 🔍

∫ 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 $x = a$.

• $f'(a) > 0$: $f$ 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 $f'$

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 $f$:

1. Zeros of $f'$ = horizontal tangent lines of $f$ (possible extrema)
2. Sign changes of $f'$ = extrema of $f$
3. Extrema of $f'$ = inflection points of $f$
4. $f'$ 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 $x=0$), it's just an inflection point.

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From $f$ to $f'$ 🔍

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

∫ Derivative Notation

Part 4 of 7 — The Language of Derivatives

1. Common Notations

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

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

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 $x$."

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Match the Notation 🔍

∫ 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: $m = f'(a)$
• Point: $(a, f(a))$

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

2. Worked Example

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

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

3. Normal Line

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

1. Find $f(a)$ — the $y$-value at the point
2. Find $f'(a)$ — the slope at the point
3. 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 🔍

∫ Problem-Solving Workshop

Part 6 of 7 — Derivative Definition Practice

Strategy: Limit-Definition Problems

When asked to find a derivative using the limit definition:

1. Write out $\frac{f(x+h)-f(x)}{h}$
2. Expand $f(x+h)$ carefully
3. Simplify — everything should cancel the $h$ in the denominator
4. 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

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

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Recognize & Evaluate 🔍

∫ 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
• Physically: instantaneous rate of change
• Algebraically: $\lim_{h \to 0} \frac{f(a+h)-f(a)}{h}$

Key Relationships

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

Differentiability Hierarchy

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

But NONE of the reverse implications hold!

Essential Formulas

• 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 $x$ near $a$

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Common AP Exam Derivative Questions

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

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Final Review 🔍