The Derivative Concept

Summary

This chapter introduces the derivative, one of the two fundamental concepts of calculus. Students will understand the derivative as a measure of instantaneous rate of change and as the slope of a tangent line. Starting with average rate of change and the difference quotient, the chapter develops the limit definition of the derivative and introduces both prime notation and Leibniz notation. After completing this chapter, students will understand what the derivative represents both graphically and in terms of rates of change.

Concepts Covered

This chapter covers the following 15 concepts from the learning graph:

Rate of Change
Average Rate of Change
Instantaneous Rate
Difference Quotient
Secant Line
Tangent Line
Slope of Tangent
Derivative Definition
Limit Definition Deriv
Derivative at a Point
Derivative Function
Derivative Notation
Prime Notation
Leibniz Notation
Differentiable Function

Prerequisites

This chapter builds on concepts from:

Introduction: The Heart of Calculus

Imagine you're watching a car's speedometer as you drive. The number you see—let's say 65 mph—tells you exactly how fast you're going at that precise moment. But how does the speedometer know? After all, speed is distance divided by time, and a single instant has no duration. This puzzle—how to measure change at an instant—is exactly what the derivative solves.

Delta Moment

"This is what I live for! I roll along curves, and my tilt at any moment IS the derivative. I don't need to travel anywhere—I just feel the slope right under my wheels."

The derivative is one of calculus's two fundamental operations (the other being integration). Where limits help us understand what happens as we approach a value, derivatives tell us precisely how fast things are changing at each instant. This chapter takes you from the intuitive idea of "rate of change" all the way to the formal definition of the derivative.

Understanding Rate of Change

Before we can talk about instantaneous rate of change, we need to understand rate of change in general. A rate of change measures how one quantity changes relative to another. You encounter rates of change constantly:

Speed: distance per unit time (miles per hour)
Growth rate: population change per year (people per year)
Slope: rise per unit run (vertical change per horizontal change)
Price change: dollars per share (stock market movements)

All these examples share a common structure: we compare how much something changes to how much something else changes.

Quantity Changing	Measured Against	Rate of Change
Distance	Time	Speed/Velocity
Temperature	Time	Heating/Cooling rate
Cost	Items produced	Marginal cost
Height	Horizontal distance	Slope
Population	Time	Growth rate

The key insight is that rate of change is always a comparison between two quantities. When we make this comparison over an interval, we get the average rate of change. When we zoom in to a single instant, we get the instantaneous rate—the derivative.

Average Rate of Change

The average rate of change of a function $f$ over an interval $[a, b]$ measures the overall change in output divided by the change in input:

\[\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}\]

This should look familiar—it's the slope formula you learned in algebra, just written with function notation.

Let's make this concrete. Suppose you drive from home to a friend's house 120 miles away, and the trip takes 2 hours. Your average speed is:

\[\frac{120 \text{ miles}}{2 \text{ hours}} = 60 \text{ mph}\]

But you weren't traveling exactly 60 mph the whole time. Sometimes you were stopped at lights, sometimes cruising at 70 on the highway. The average rate of change tells us the overall rate, not what happened at any specific moment.

Diagram: Average Rate of Change Explorer

Average Rate of Change Explorer MicroSim

Type: microsim

Purpose: Allow students to explore average rate of change by moving two points along a curve and seeing the secant line and slope calculation update in real-time.

Learning Objective: Students will apply the average rate of change formula to various functions (Bloom Level 3: Apply)

Bloom Taxonomy Verb: calculate, demonstrate

Visual elements:

Coordinate plane with gridlines (-10 to 10 on both axes)
A curve representing a function (selectable: parabola, cubic, sine)
Two draggable points A and B on the curve
Secant line connecting points A and B
Dynamic display showing: coordinates of A and B, calculation of slope

Interactive controls:

Dropdown to select function: $f(x) = x^2$, $f(x) = x^3 - 3x$, $f(x) = \sin(x)$
Two draggable points on the curve
Reset button

Behavior:

As user drags points, secant line updates
Slope calculation displayed as: $\frac{f(b) - f(a)}{b - a} = \frac{[value] - [value]}{[value] - [value]} = [result]$
Color coding: rise in blue vertical segment, run in red horizontal segment

Instructional Rationale: Direct manipulation of points helps students connect the formula to geometric interpretation. Seeing the calculation update in real-time builds procedural fluency while maintaining conceptual understanding.

Implementation: p5.js with draggable points

Example: Calculating Average Rate of Change

For $f(x) = x^2$, find the average rate of change from $x = 1$ to $x = 4$.

\[\text{Average rate of change} = \frac{f(4) - f(1)}{4 - 1} = \frac{16 - 1}{3} = \frac{15}{3} = 5\]

This tells us that on average, the function increases by 5 units for each unit increase in $x$ over this interval. But at $x = 1$, the curve is relatively flat, while at $x = 4$, it's much steeper. The average doesn't tell us what's happening at any specific point.

The Secant Line Connection

When you plot two points on a curve and draw a line through them, that line is called a secant line. The slope of the secant line is the average rate of change between those two points.

The word "secant" comes from the Latin secare, meaning "to cut." A secant line cuts through the curve at two points, unlike a tangent line which just touches it at one point.

Key Connection

Average rate of change over $[a, b]$ = Slope of secant line through $(a, f(a))$ and $(b, f(b))$

This geometric interpretation is crucial. Every time you calculate an average rate of change, you're finding the slope of a secant line. And when we want to find the instantaneous rate, we'll see what happens to these secant lines as the two points get closer together.

The Difference Quotient

The difference quotient is a specific way to write the average rate of change that sets us up perfectly for taking a limit. Instead of using two separate points $a$ and $b$, we use a single starting point $x$ and add a small increment $h$:

\[\frac{f(x + h) - f(x)}{h}\]

Here's what each part means:

$x$ is our starting point
$h$ is the horizontal distance to our second point (can be positive or negative)
$f(x)$ is the function value at the starting point
$f(x + h)$ is the function value at the second point
The whole expression is the slope of the secant line between $(x, f(x))$ and $(x+h, f(x+h))$

The difference quotient is powerful because it lets us ask: "What happens to the slope as $h$ gets smaller and smaller?"

Example: Finding the Difference Quotient for $f(x) = x^2$

\[\frac{f(x + h) - f(x)}{h} = \frac{(x + h)^2 - x^2}{h}\]

Let's expand and simplify:

\[= \frac{x^2 + 2xh + h^2 - x^2}{h} = \frac{2xh + h^2}{h} = \frac{h(2x + h)}{h} = 2x + h\]

This simplified form, $2x + h$, tells us the slope of any secant line starting at $x$ with horizontal distance $h$. Notice that as $h$ gets closer to 0, this expression approaches $2x$.

From Average to Instantaneous: The Tangent Line

Here's the key question: Can we find the rate of change at a single point?

At first, this seems impossible. Rate of change requires two points—you need a change in both the input and output. But calculus shows us a way: we look at what happens to the average rate of change as the two points get infinitely close together.

Delta Moment

"Imagine I'm sitting at a single point on a curve. Which way am I tilted? That direction—the slope of the ground right beneath my wheels—that's what we're after. The tangent line shows exactly which way I'm pointing!"

A tangent line is a line that just "touches" a curve at a single point, matching the curve's direction at that point. Unlike a secant line that crosses through the curve, a tangent line grazes it.

The slope of the tangent line represents the instantaneous rate of change—the rate of change at a single instant, at a single point.

Diagram: Secant Lines Approaching Tangent Line

Secant Lines Approaching Tangent Line MicroSim

Type: microsim

Purpose: Visualize how secant lines approach the tangent line as the second point moves closer to the fixed point.

Learning Objective: Students will explain how the limit of secant line slopes gives the tangent line slope (Bloom Level 2: Understand)

Bloom Taxonomy Verb: explain, interpret

Visual elements:

Coordinate plane with a parabola $f(x) = x^2$
Fixed point P at chosen x-value (default $x = 1$)
Moving point Q along the curve
Secant line PQ that updates as Q moves
Ghost tangent line showing the target slope
Table showing h values and corresponding slopes

Interactive controls:

Slider to control position of Q (equivalently, the value of h)
Slider to choose the fixed point P's x-coordinate
Animation button: "Watch h approach 0"
Display: current h value, current slope, limit slope

Data Visibility Requirements:

Stage 1: Show secant line with h = 2, display slope value
Stage 2: As slider moves, show h = 1, h = 0.5, h = 0.1, h = 0.01
Stage 3: Table fills with (h, slope) pairs
Stage 4: Secant line visually merges with tangent line
Final: Display "As h → 0, slope → [tangent slope]"

Behavior:

As h decreases, Q slides toward P
Secant line rotates to approach tangent line
Numerical display shows slope converging
At very small h, secant and tangent lines are visually indistinguishable

Instructional Rationale: Step-through visualization with concrete numerical values helps students understand the limiting process. Seeing the secant line physically rotate to become the tangent line creates a lasting mental image.

Implementation: p5.js with animated slider

The Limit Definition of the Derivative

Now we can state the formal definition. The derivative of $f$ at $x = a$ is defined as:

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

This definition says: take the difference quotient (the slope of the secant line) and see what happens as $h$ approaches 0. If this limit exists, that's the derivative at $x = a$.

Equivalently, using the points $(a, f(a))$ and $(x, f(x))$:

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

Both forms say the same thing: the derivative is the limit of the average rate of change as the interval shrinks to nothing.

Why This Works

When $h = 0$, the difference quotient is $\frac{0}{0}$—undefined. But the limit asks what the quotient approaches as $h$ gets close to 0, not what happens when $h$ equals 0. This is exactly the situation limits were designed for!

Derivative at a Point vs. Derivative Function

There's an important distinction between finding the derivative at a specific point and finding the derivative as a function.

The derivative at a point $f'(a)$ is a single number—the slope of the tangent line at $x = a$.

The derivative function $f'(x)$ is a new function that gives you the derivative at any point where it exists. We find it by keeping $x$ as a variable throughout the limit process.

Example: Finding the Derivative Function for $f(x) = x^2$

We already found that the difference quotient simplifies to $2x + h$. Now take the limit:

\[f'(x) = \lim_{h \to 0} (2x + h) = 2x\]

So the derivative function is $f'(x) = 2x$. This tells us:

At $x = 1$: slope is $f'(1) = 2(1) = 2$
At $x = 3$: slope is $f'(3) = 2(3) = 6$
At $x = -2$: slope is $f'(-2) = 2(-2) = -4$

The derivative function captures all the instantaneous rates of change in one formula.

Diagram: Function and Derivative Side by Side

Function and Derivative Comparison MicroSim

Type: microsim

Purpose: Show a function and its derivative side by side, with a synchronized point that shows the connection between function value, tangent slope, and derivative value.

Learning Objective: Students will analyze the relationship between a function graph and its derivative graph (Bloom Level 4: Analyze)

Bloom Taxonomy Verb: analyze, examine, compare

Visual elements:

Two coordinate planes side by side
Left panel: Original function $f(x)$ with a point and tangent line
Right panel: Derivative function $f'(x)$ with corresponding point
Vertical dashed line connecting the x-positions in both panels
Tangent line on left, horizontal line showing height on right

Interactive controls:

Horizontal slider to move the point along the x-axis
Dropdown to select function: $x^2$, $x^3$, $\sin(x)$, $e^x$
Toggle: Show/hide tangent line
Display: Current x, $f(x)$, $f'(x)$

Behavior:

Moving slider updates both panels simultaneously
Tangent line slope on left matches the height of point on right
When tangent is horizontal (slope = 0), derivative point is at y = 0
When tangent slopes up, derivative is positive; down means negative

Instructional Rationale: Simultaneous visualization of function and derivative builds understanding of their relationship. Moving a synchronized point helps students see that the derivative value equals the tangent slope.

Implementation: p5.js with two canvases or split canvas

Derivative Notation

Mathematicians use several different notations for derivatives. Each has its advantages depending on the context.

Prime Notation (Lagrange Notation)

The most common notation uses a prime symbol:

$f'(x)$ — "f prime of x" — the derivative of $f$ with respect to $x$
$f'(a)$ — "f prime of a" — the derivative evaluated at $x = a$
$y'$ — "y prime" — the derivative when $y$ is the dependent variable

This notation is compact and easy to write, making it popular for calculations.

Leibniz Notation

Developed by Leibniz, this notation emphasizes that the derivative is a ratio of infinitesimal changes:

$\frac{dy}{dx}$ — "dy dx" or "the derivative of y with respect to x"
$\frac{df}{dx}$ — the derivative of function $f$ with respect to $x$
$\frac{d}{dx}[f(x)]$ — "d dx of f of x" — an operator notation

Leibniz notation is powerful because it reminds us that the derivative came from a ratio (the difference quotient). It's especially useful in applications like related rates and the chain rule.

Notation Caution

The expression $\frac{dy}{dx}$ looks like a fraction, and sometimes it behaves like one. But technically it's a single symbol representing the derivative, not an actual fraction. Be careful when manipulating it algebraically.

Other Notations

You may also see:

$Df$ or $D_x f$ — operator notation (mostly in advanced courses)
$\dot{y}$ — Newton's notation (a dot above the variable, common in physics for time derivatives)

Notation	Read as	When to use
$f'(x)$	"f prime of x"	General calculations
$\frac{dy}{dx}$	"dy dx"	When variables are named, chain rule, related rates
$\frac{d}{dx}[expression]$	"d dx of..."	When differentiating an expression
$\dot{y}$	"y dot"	Physics, time derivatives

What Makes a Function Differentiable?

A function is differentiable at a point if the derivative exists at that point—meaning the limit that defines the derivative exists and is finite.

A function is differentiable on an interval if it's differentiable at every point in that interval.

For the limit $\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$ to exist, the function must:

Be defined at $x = a$ (so $f(a)$ exists)
Be defined near $x = a$ (so we can approach from both sides)
Have the left-hand and right-hand limits of the difference quotient be equal

When a function is differentiable at a point, the curve is "smooth" there—it has a well-defined tangent line. We'll explore the situations where functions aren't differentiable in the next chapter.

Delta Moment

"When I'm on a smooth curve, I know exactly which way I'm pointing—my tilt is well-defined. But at a sharp corner? I'd have to pick a direction, and there's no right answer. That's a non-differentiable point!"

Interpreting the Derivative

The derivative has two fundamental interpretations that you should always keep in mind:

Geometric Interpretation: The derivative $f'(a)$ is the slope of the tangent line to the graph of $f$ at $x = a$.

Rate Interpretation: The derivative $f'(a)$ is the instantaneous rate of change of $f$ with respect to $x$ at $x = a$.

These two interpretations are different ways of describing the same mathematical object. Depending on the context, one interpretation may be more useful than the other:

In physics problems about motion, think "rate of change"
In graphing problems, think "slope of tangent"
In optimization problems, both interpretations matter

Diagram: Derivative Interpretation Selector

Derivative Interpretation Selector MicroSim

Type: microsim

Purpose: Present the same derivative in both graphical (slope) and contextual (rate) interpretations, showing they're two views of the same concept.

Learning Objective: Students will interpret the derivative both as slope and as rate of change in context (Bloom Level 2: Understand)

Bloom Taxonomy Verb: interpret, explain

Visual elements:

Left panel: Graph with tangent line showing slope interpretation
Right panel: Contextual display (speedometer, growth meter, etc.)
Central display: The derivative value with units
Toggle between contexts: distance/time (velocity), population/time (growth), cost/units (marginal cost)

Interactive controls:

Dropdown: Select context (Car trip, Population, Manufacturing cost)
Slider: Move along the x-axis to see derivative at different points
Toggle: Switch between graphical and contextual emphasis

Behavior:

Changing context updates the labels and units but not the underlying function
Same derivative value shown as "slope = 3" and "velocity = 3 m/s"
Emphasizes that the mathematics is identical regardless of interpretation

Instructional Rationale: Seeing the same derivative in multiple contexts builds transfer and deepens understanding that the derivative is a single concept with multiple applications.

Implementation: p5.js with context-switching panels

Computing Derivatives Using the Limit Definition

Let's practice finding derivatives using the limit definition. This process has three steps:

Write the difference quotient $\frac{f(x+h) - f(x)}{h}$
Simplify algebraically (the goal is to cancel $h$ from the denominator)
Take the limit as $h \to 0$

Example 1: Find the derivative of $f(x) = 3x + 5$

Step 1: Difference quotient

\[\frac{f(x+h) - f(x)}{h} = \frac{[3(x+h) + 5] - [3x + 5]}{h}\]

Step 2: Simplify

\[= \frac{3x + 3h + 5 - 3x - 5}{h} = \frac{3h}{h} = 3\]

Step 3: Take the limit

\[f'(x) = \lim_{h \to 0} 3 = 3\]

The derivative of a linear function is its slope—exactly what we'd expect!

Example 2: Find the derivative of $f(x) = x^3$

Step 1: Difference quotient

\[\frac{f(x+h) - f(x)}{h} = \frac{(x+h)^3 - x^3}{h}\]

Step 2: Expand using $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$

\[= \frac{x^3 + 3x^2h + 3xh^2 + h^3 - x^3}{h} = \frac{3x^2h + 3xh^2 + h^3}{h}\]

Factor out $h$:

\[= \frac{h(3x^2 + 3xh + h^2)}{h} = 3x^2 + 3xh + h^2\]

Step 3: Take the limit

\[f'(x) = \lim_{h \to 0} (3x^2 + 3xh + h^2) = 3x^2\]

Example 3: Find the derivative of $f(x) = \frac{1}{x}$

Step 1: Difference quotient

\[\frac{f(x+h) - f(x)}{h} = \frac{\frac{1}{x+h} - \frac{1}{x}}{h}\]

Step 2: Find common denominator in numerator

\[= \frac{\frac{x - (x+h)}{x(x+h)}}{h} = \frac{\frac{-h}{x(x+h)}}{h} = \frac{-h}{h \cdot x(x+h)} = \frac{-1}{x(x+h)}\]

Step 3: Take the limit

\[f'(x) = \lim_{h \to 0} \frac{-1}{x(x+h)} = \frac{-1}{x \cdot x} = -\frac{1}{x^2}\]

The Tangent Line Equation

Once you know the derivative at a point, you can write the equation of the tangent line. Using point-slope form:

\[y - f(a) = f'(a)(x - a)\]

Or equivalently:

\[y = f(a) + f'(a)(x - a)\]

Example: Tangent line to $f(x) = x^2$ at $x = 3$

We need two things: the point and the slope.

Point: $(3, f(3)) = (3, 9)$
Slope: $f'(3) = 2(3) = 6$ (using $f'(x) = 2x$ from earlier)

Tangent line equation:

\[y - 9 = 6(x - 3)$$ $$y = 6x - 18 + 9$$ $$y = 6x - 9\]

Diagram: Tangent Line Calculator

Tangent Line Calculator MicroSim

Type: microsim

Purpose: Given a function and a point, show the step-by-step calculation of the tangent line equation.

Learning Objective: Students will apply the derivative to write tangent line equations (Bloom Level 3: Apply)

Bloom Taxonomy Verb: calculate, apply, solve

Visual elements:

Coordinate plane showing the function
Highlighted point of tangency
Tangent line drawn through the point
Step-by-step calculation panel showing:
Point: $(a, f(a)) = ...$
Slope: $f'(a) = ...$
Equation: $y - f(a) = f'(a)(x - a)$
Simplified: $y = mx + b$

Interactive controls:

Input field or slider to choose x-coordinate of tangent point
Dropdown: Select function from preset list
Button: "Show Steps" to reveal calculation
Toggle: Show/hide the tangent line

Behavior:

User selects point, simulation calculates and displays tangent line
Step-by-step work shown in calculation panel
Tangent line updates dynamically as user changes input

Instructional Rationale: Seeing the complete calculation process alongside the geometric result reinforces the connection between symbolic and graphical representations.

Implementation: p5.js with dynamic text rendering

Summary: The Big Picture

Let's step back and see how all these concepts connect:

Rate of Change
     ↓
Average Rate of Change = (f(b) - f(a))/(b - a) = Slope of Secant Line
     ↓
Difference Quotient = (f(x+h) - f(x))/h = Slope of Secant (variable form)
     ↓
Take Limit as h → 0
     ↓
DERIVATIVE = f'(x) = lim[h→0] (f(x+h) - f(x))/h = Slope of Tangent Line
     ↓
= Instantaneous Rate of Change

The derivative transforms the intuitive idea of "rate of change" into a precise mathematical concept. It answers the question: "How fast is the function changing at this exact point?"

Delta's Pun Corner

"Why do I love derivatives? Because they give my life direction! Without them, I'd just be a triangle with no sense of slope."

Key Takeaways

Average rate of change measures overall change over an interval: $\frac{f(b) - f(a)}{b - a}$
The difference quotient $\frac{f(x+h) - f(x)}{h}$ represents the slope of a secant line
The derivative is the limit of the difference quotient as $h \to 0$: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$
The derivative has two key interpretations:
Geometric: Slope of the tangent line
Physical: Instantaneous rate of change
Prime notation ($f'(x)$) and Leibniz notation ($\frac{dy}{dx}$) both represent the derivative
A function is differentiable at a point if the derivative exists there
To find a tangent line: use point-slope form $y - f(a) = f'(a)(x - a)$

Check Your Understanding: What is the derivative of $f(x) = x^2 - 4x + 7$ at $x = 2$?

First, find $f'(x)$ using the limit definition or known rules:

$f'(x) = 2x - 4$

Then evaluate at $x = 2$:

$f'(2) = 2(2) - 4 = 0$

The derivative is 0 at $x = 2$, which means the tangent line is horizontal there. This point is a critical point of the function!

See Annotated References

Notation	Read as	When to use
\(f'(x)\)	"f prime of x"	General calculations
\(\frac{dy}{dx}\)	"dy dx"	When variables are named, chain rule, related rates
\(\frac{d}{dx}[expression]\)	"d dx of..."	When differentiating an expression
\(\dot{y}\)	"y dot"	Physics, time derivatives