Intuition The ONE core ideaA derivative is the steepness of a curve at a single point , found by measuring the slope between two points and then sliding them together until they merge. Everything below is the vocabulary and pictures you need so that when the parent note writes lim h → 0 h f ( x + h ) − f ( x )
This page assumes you know nothing beyond "a graph is a picture of numbers." We will earn every letter, one at a time, and each new idea leans on the one before it.
f f machine : you feed it a number, it gives back one number. We write f ( x ) x x input slot .
f x f ( x ) output ).
Think of a vending machine. Press button x = 3 f ( 3 ) = 9 f ( x ) = x 2 rule is fixed; only your input changes.
Why the topic needs it: the derivative is a fact about a function . No machine, nothing to differentiate.
f x f ( x ) point ( x , f ( x ) ) x f ( x ) x curve . That curve is the graph.
The horizontal axis measures the input x
The vertical axis measures the output f ( x )
A single point on the curve is written ( x , f ( x ) )
Intuition Why a curve, not a line?If bigger inputs give proportionally bigger outputs, you get a straight line. But f ( x ) = x 2 faster and faster , so the picture bends upward — a curve. Different places on that curve tilt by different amounts, and that tilt is the whole story of this topic .
Why the topic needs it: "slope" and "steepness" are things you see on the graph. Without the picture there is nothing to measure the steepness of.
Definition Slope of a straight lineSlope answers: "for every step I take sideways, how much do I climb?" Pick two points on the line. The run is how far right you moved; the rise is how far up you climbed. Then
slope = run rise = change in sideways change in height
Run = change in the input (horizontal).Rise = change in the output (vertical).A big slope = steep. A slope of 0 negative slope = going downhill as you move right.
Intuition Why "over"? Why divide?Dividing rise by run gives climb per single sideways step . That makes slopes comparable: a staircase that climbs 2 m over 4 m (2 1 3
Common mistake Slope needs TWO pointsYou cannot compute rise-or-run from a single dot — a change needs a "before" and an "after." This single fact is the entire reason the topic exists: a curve's slope changes at every point, yet slope seems to need two points. The trick (Section 6) resolves this.
Why the topic needs it: the derivative is a slope. Every worked example in the parent ends with a slope value.
See Average vs Instantaneous Rate of Change — a two-point slope is an average rate; the derivative is the instantaneous one.
h h small distance we step to the right along the input axis. Starting at input x x + h h 0.1 0.001
x x + h buddy point a little to the right.h run , in the language of Section 3).
Intuition Picture the buddyImagine your finger on the curve at x x + h h
h h alive (non-zero) while we do algebra. Only at the very last step do we ask what happens as it heads to zero . Setting h = 0
Why the topic needs it: h
f ( x + h ) x + h f ( x + h ) height of the curve at the buddy point . You get it by replacing every x ( x + h )
Worked example Concrete substitutionIf f ( x ) = x 2 f ( x + h ) = ( x + h ) 2 ( x + h ) 2 = x 2 + 2 x h + h 2 f ( x ) = x 1 f ( x + h ) = x + h 1
Common mistake Substitute EVERYWHEREFor f ( x ) = x 2 ( x + h ) 2 not x 2 + h x + h
Why the topic needs it: to measure a slope you need the height at both points — f ( x ) f ( x + h )
Now we combine Sections 3, 4, 5. The two points are:
Point A: address ( x , f ( x ) )
Point B: address ( x + h , f ( x + h ) )
The rise (climb from A to B) is f ( x + h ) − f ( x ) run (sideways from A to B) is ( x + h ) − x = h
A secant is any straight line that cuts through a curve at two points. Its slope is the average steepness of the curve between those two points.
Why the topic needs it: this is the object the whole topic is built on. The derivative is what this quantity approaches as the buddy slides in. See Secant and Tangent lines .
lim h → 0 ( ⋯ ) "the value the expression settles on as h 0 0 The arrow → destination , not the endpoint.
Intuition Why we need a limit hereAs the buddy point slides toward your point (h → 0 rotates and, in the merged position, just touches the curve — that touching line is the tangent . But we can't land exactly on h = 0 heading . The limit is the tool that answers "heading toward what?" — exactly the question this topic needs and no other tool answers it. See Limits — formal definition and one-sided limits .
The tangent touches the curve at one point and matches its steepness there. Its slope is the derivative. As h → 0 →
Why the topic needs it: without a limit, "slope at a single point" is impossible — the run would be 0
If we set h = 0 before simplifying, the run becomes 0 f ( x ) − f ( x ) = 0 0 0
0 0 0 0 0 indeterminate form — undecided until you do more work. See Indeterminate forms 0 over 0 .
Intuition The escape routeFirst do algebra to cancel the troublesome h h → 0
Why the topic needs it: it explains the strict order — simplify first, take the limit second — that makes the whole method work.
f ′ ( x ) ′ prime ) means "the derivative of." So f ′ ( x ) f x the slope of f x . It is itself a new machine: input a point, output the steepness there.
f ′ f f ( 10 ) = 100 f ′ ( 10 ) = 20 f ( x ) = x 2 height , the other a steepness . Never read f ′ f f Continuity and Differentiability .
Why the topic needs it: f ′ ( x ) answer the topic produces — the derivative as a function of position.
Now the parent note's headline makes sense with no strangers left:
f ′ ( x ) = lim h → 0 h f ( x + h ) − f ( x )
Read it as a sentence: "The steepness of f x f ′ ( x ) the fraction ) heads toward (lim h → 0
Graph turns machine into a curve
h is a tiny step to x plus h
f of x plus h is height at buddy
Difference quotient secant slope
Zero over zero is forbidden
Test yourself — cover the right side and answer out loud.
What does f ( x ) The number a machine f x
On a graph, what does the point ( x , f ( x ) ) Go across by x f ( x )
Define slope without symbols. Rise over run — how much you climb for each sideways step.
What is h A tiny sideways distance to the buddy point x + h non-zero while we do algebra.
How do you compute f ( x + h ) f ( x ) = x 2 Replace every x ( x + h ) ( x + h ) 2 = x 2 + 2 x h + h 2
What geometric object does h f ( x + h ) − f ( x ) The slope of the secant line through the two curve points.
What does lim h → 0 What value the expression settles on as h 0 0
Why is setting h = 0 The run becomes 0 0 0
What does f ′ ( x ) The slope of f x
As h → 0 tangent
