Free lesson · Differential calculus

Start with a straight line: input, output and slope

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Start with the idea

First calculate outputs.
Then compare two outputs.
A slope is output change divided by input change.

Symbols, units & horizon

x: starting input, dimensionless here
f(x): output, dimensionless here
h: nonzero input change
f(x+h): output after that change
/: divide the output change by the input change

When and why to use this

Use a slope to understand a sensitivity. For a fixed holding, value changes linearly with price; the slope is the quantity held.

Function: a rule that maps an input to an output.
Read f(x) as “the output of f at input x.”
Use $f (x) = 2 x + 1$ : multiply x by 2, then add 1.
Input 1 gives output 3. Input 2 gives output 5.
Slope: rise divided by run.
A straight line has the same slope over every nonzero interval.

f (x) = 2 x + 1, \frac{f (x + h) - f (x)}{h} = 2 (h \neq 0)

Core rule · definition and worked arithmetic

Start with a straight line: input, output and slope

Evaluate the shifted input: $f (x + h) = 2 x + 2 h + 1$ .
Subtract the original output: $f (x + h) - f (x) = 2 h$ .
Divide by the nonzero input change: $\frac{2 h}{h} = 2$ .

Work it by hand

From x=1 to x=2, slope=(5−3)/(2−1)=2. From x=1 to x=1.1, slope=(3.2−3)/.1=2.

Use the rule

Name the inputs and units.
Work the small example by hand.
Check the result before continuing to the next lesson.

Before moving on

Explain the core rule in one sentence, reproduce the worked calculation and solve both practice variations.

Further reading: OpenStax · Differentiation rules · textbook checked 12 September 2026 ↗

Research sources, review dates and limitations

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

def line(x):
    return 2*x+1

def line_slope(x, h):
    if h == 0: raise ValueError("Nonzero input change required")
    return (line(x+h)-line(x))/h

print(line(1), line(2), line_slope(1,1))

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