Derivatives from Scratch: Slopes, Graphs, and the Idea of Instantaneous Rate

1) Slope as “Rate of Change” on a Graph

When you graph a function y = f(x), each point (x, y) represents an input and its output. The most important visual idea behind derivatives is this: the derivative measures how steep the graph is, which is the same as how fast the output changes as the input changes.

On a graph, “steepness” is captured by slope. A positive slope means the graph goes up as you move right; a negative slope means it goes down.

Units matter: “output units per input unit”

Slope is not just a number; it carries units. If x is measured in hours and y is measured in degrees Celsius, then slope is measured in °C per hour. If x is seconds and y is meters, slope is m/s.

• Interpretation template: “y changes by (slope) units for each 1 unit increase in x.”

2) Average Rate of Change: Secant Lines

Before talking about an “instantaneous” rate, start with an average rate over an interval. Pick two points on the graph: (x1, f(x1)) and (x2, f(x2)). The line through them is a secant line. Its slope is the average rate of change of the function from x1 to x2.

How to compute the slope from two points

The slope of the secant line is:

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average rate of change = (f(x2) - f(x1)) / (x2 - x1)

This is often written as:

m = Δy / Δx

where Δy = f(x2) - f(x1) and Δx = x2 - x1.

Step-by-step example (with units)

Suppose a room’s temperature is modeled by T(t), where t is time in hours and T is in °C. You measure:

• T(1) = 18°C
• T(4) = 24°C

Compute the average rate of change from t=1 to t=4:

(T(4) - T(1)) / (4 - 1) = (24 - 18) / 3 = 2 °C per hour

Meaning: Over that 3-hour interval, the temperature increased on average by 2°C each hour.

Reflection prompt (write a sentence)

• Turn 2 °C per hour into a contextual sentence: “From hour 1 to hour 4, the temperature is rising at an average rate of 2°C per hour.”

3) Estimating Slopes from Graphs (Before Calculating)

Often you can estimate slope by “rise over run” directly from a graph grid. The key is to pick two clear points on the curve (or on a line) and count vertical and horizontal changes.

Mini-exercise A: Estimate then compute

Look at this simple line graph (ASCII sketch):

y (units)  8 |            • (4,8)           6 |        • (2,6)               4 |    • (1,4)                   2 |• (0,2)                       0 +--------------------------- x (units)    0    1    2    3    4

• Estimate: From (0,2) to (4,8), the rise is about 6 and the run is 4, so slope ≈ 6/4 = 1.5.
• Compute exactly: (8 - 2)/(4 - 0) = 6/4 = 1.5.

Reflection prompt

• If x is hours and y is liters, write: “The amount is increasing by 1.5 liters per hour.”

4) Why Average Rate Isn’t Enough: The Need for Instantaneous Rate

Average rate of change is useful, but many real situations change speed over time. For example, a car may speed up, slow down, and stop. If you compute average speed over 10 seconds, you do not know the speed at exactly 3 seconds.

Motion example: changing speed

Let s(t) be position (meters) at time t (seconds). The average velocity from t=2 to t=5 is:

(s(5) - s(2)) / (5 - 2)  (meters per second)

But if the object accelerates, the velocity at t=3 could be very different from that average.

This motivates the central idea: instantaneous rate of change is the rate “right now,” at a single input value. Graphically, it corresponds to the slope of the tangent line—the line that just touches the curve at a point and matches its local direction.

5) From Secant to Tangent: Shrinking the Interval

To approximate the instantaneous rate at x=a, compute slopes of secant lines using points closer and closer to a.

Pick a nearby point a+h. The secant slope is:

(f(a+h) - f(a)) / ((a+h) - a) = (f(a+h) - f(a)) / h

As h gets smaller (approaching 0), the secant line “rotates” toward the tangent line. The instantaneous rate at a is the limiting value of these slopes.

Numerical approximation example (table of secant slopes)

Suppose f(x) = x^2 and you want the instantaneous rate at a=2. Compute secant slopes for smaller h:

The slopes approach 4. That means the instantaneous rate of change of x^2 at x=2 is about 4 (and in later chapters you’ll formalize this as the derivative).

Reflection prompt

• If f measured distance in meters and x measured time in seconds, what would “instantaneous rate ≈ 4” mean? Write a sentence like: “At t=2 seconds, the object’s speed is about 4 meters per second.”

6) Reading Increasing/Decreasing vs. Steep/Flat from Graphs

Two related but different questions:

• Increasing vs. decreasing: Is the function going up or down as x increases?
• Steep vs. flat: How large is the slope (in magnitude)?

A graph can be increasing but nearly flat (small positive slope), or increasing and very steep (large positive slope). Similarly, it can be decreasing gently (small negative slope) or decreasing sharply (large negative slope).

Graph comparison 1: same direction, different steepness

Imagine two hills on a map: both go uphill (increasing), but one is a gentle incline and the other is a steep climb. On a function graph, that difference is captured by the size of the slope.

Graph A (gentle increase):      Graph B (steep increase): y                           y |        ____               |              / |      /                    |            /  |    /                      |          /    |___/                       |________/________ x          x

• Graph A: slope is positive but small (nearly flat).
• Graph B: slope is positive and large (very steep).

Mini-exercise B: Estimate slopes at different points

Consider this curve sketch (not to scale):

y |                 • C |              _/   |           _/      |        _/         |     _/            |___/_______________ x      A   B

Estimate the slope (instantaneous rate) at points A, B, and C by imagining a tiny tangent line at each point:

• At A (near the bottom left), is the slope small or large? Positive or negative?
• At B (middle), does the graph look steeper than at A?
• At C (upper right), does the graph start flattening or keep getting steeper?

Then describe your estimates using words like “about 0,” “small positive,” “large positive.”

Graph comparison 2: increasing vs. decreasing

Increasing:                 Decreasing: y                           y |      /                    |\ |    /                      | \ |  /                        |  \ |_/________________ x        |___\___________ x

• Increasing graph: slopes are positive.
• Decreasing graph: slopes are negative.

Mini-exercise C: Sign of slope and meaning

For each situation, decide whether the slope should be positive, negative, or zero:

• Altitude vs. time while an elevator goes up.
• Fuel in a tank vs. time while driving.
• Distance remaining to a destination vs. time while walking toward it.
• Temperature vs. time while a thermostat holds steady.

Then write one sentence for each using “per” units (for example, “Fuel is decreasing by about 0.3 liters per minute”).

7) Turning Slope Values into Context Sentences

A slope value becomes meaningful when you translate it into a statement about change.

Examples of slope interpretations

• Slope = 2 °C/hour: “Temperature is rising 2°C per hour.”
• Slope = −5 m/s: “Velocity is decreasing by 5 meters per second each second” (negative slope indicates a downward trend in the graphed quantity).
• Slope = 0: “The quantity is staying constant at that moment (flat tangent).”

Mini-exercise D: Translate and check units

Match each slope to a correct sentence (and fix any unit mistakes):

• 1) Slope = −3, where y is dollars and x is days.
• 2) Slope = 0.8, where y is kilometers and x is hours.
• 3) Slope = 12, where y is milliliters and x is seconds.

Write your answers in the form: “y is (increasing/decreasing) by ___ (output units) per (input unit).”