Chapter 1 · The speedometer problem · ~4 min
A derivative is a speedometer. You have used one since you were four.
Your GPS knows where you are at every moment. Boring. The speedometer answers a much stranger question: how fast are you going right now? Not on average. Not over the last minute. Now.
Here is why that question is strange. Speed is distance divided by time. But "right now" contains zero distance and zero time. So your speed right now is, apparently, 0 ÷ 0.
0 ÷ 0 is not a number. It is a cry for help. Division would like to speak to a lawyer. For two thousand years this dead end is roughly where things stood.
Newton's move (and Leibniz's, independently, hence a legendary academic feud): stop asking about the instant. Ask about a small window around the instant. Then shrink the window, and watch where the answers are heading. If they all crowd toward one number, name that number the answer.
Predict first · it locks the lesson in
The car's position is s(t) = t² meters. At t = 2 s, as the window shrinks, the average speed heads toward…
what just happened
You never divided by zero. You watched a trend and named its destination. That naming trick is called a limit, and it is the only genuinely new idea in this entire subject. Everything else in calculus is this trick wearing different outfits.
The result gets written ds/dt: tiny change in position over tiny change in time. It is not scary. It is a speedometer.
Chapter 2 · The zoom trick · ~2 min
Every smooth curve is secretly straight. Zoom in and it confesses.
The Earth is a sphere. Your kitchen floor is flat. Both are true, because you live very, very zoomed in.
Curves work the same way. Keep zooming into any smooth curve and the wiggle gives up. What remains is a straight line, and straight lines have a slope: one honest number. That number is the derivative at that point.
what just happened
This is the whole hustle. Curves are hard and lines are easy, so we zoom until the curve becomes a line, then read off the slope. A semester of Calculus 1 is this move, repeated with increasingly dramatic functions.
Chapter 3 · Slope, everywhere, all at once · ~4 min
Take the slope at every point and you get a whole new curve.
Now stop thinking of the derivative as one number. Ask "what is the slope here?" at every single point, and plot the answers. You get a second curve: the full slope report of the first one.
You are going to draw one. By hand. Like a Victorian mathematician with a trackpad. Scrub the dot along the hill. Your trail below is the derivative.
drag the dot. go on.
what just happened
The curve you painted is x² − 1, the derivative of x³/3 − x. Notice what your hands learned: on a hilltop the slope crosses zero. On the steepest stretch it peaks. The derivative is a running commentary on the original.
In class you would now memorize shortcut rules (x² becomes 2x, and so on). That is all the rules are: cached answers to the game you just played manually. The insight was the game, not the cache.
Interlude · The costume department · 1 min
The notation is Victorian emoji.
| dx | "dee-ex" | A ludicrously tiny step in x. That's it. That's the whole symbol. |
| dy/dx | "dee-y, dee-x" | Tiny rise over tiny run. A slope. A speedometer. Chapter 1, formalized. |
| ∫ | "the swoopy S" | Literally an S, for Sum. Leibniz stretched it vertically because the man had flair. |
| ∫ f(x) dx | "the whole outfit" | "Add up f × tiny-step, for a million tiny steps." A sum of slivers. You'll build one in the next chapter. |
notation is the costume department, not the plot.
Chapter 4 · The odometer problem · ~3 min
An integral is an odometer: a million tiny movements, added up.
Now reverse the con. Your speedometer works fine, but the GPS is dead. You know your speed at every instant. Question: how far did you actually travel?
If speed were constant, this is primary school: distance = speed × time. But speed keeps changing. So we cheat: chop time into slices, pretend speed is constant inside each slice, multiply, and add up all the slices. The answer is wrong. But it gets less wrong as the slices get thinner. You can see where this is going.
Predict first
Rectangles tucked under a wiggly speed curve will…
what just happened
Recognize the move? Shrink, watch the trend, name the destination. Same trick as Chapter 1, pointed the other way. The ∫ sign is an S for "sum": infinitely many, infinitely thin rectangles. The limit is what stops that sentence from being nonsense.
The truth, 18.0398…, comes from a formula we'll meet in about ninety seconds. The rectangles never quite get there. They just run out of excuses.
Chapter 5 · The plot twist · ~3 min
The speedometer and the odometer are the same instrument, read backwards.
Chapter 1: position → slope → speed. Chapter 4: speed → area → position. Two problems, solved centuries apart, by rival geniuses, for different reasons… and they are perfect inverses. The derivative undoes the integral. This is the Fundamental Theorem of Calculus, and it is one of the great plot twists in human thought.
Don't take my word for it. Scrub the timeline and watch two numbers: how fast the shaded area is growing, and the height of the curve. Everywhere you stop, at every single moment, they match.
rate = height. scrub anywhere. it never misses.
That's it. That's calculus.
"The area grows at the rate of the height." That sentence is a full semester. You now own it. Everything after this in a calculus course is technique: rules, tricks, special cases, exams designed by people who peaked early.
One more thing. Your odometer reads 18.04 at t = 6. Scroll up: that is the exact number Chapter 4's rectangles were crawling toward. Nothing in this subject is a coincidence. somewhere, the math police weep with joy.
Bonus level · One more dimension · ~3 min
3D calculus is the same trick, one direction at a time.
Swap the curve for a surface: a landscape with hills. Standing at a point, "slope" is no longer one number, because it depends which way you walk. Facing east and facing north give different steepness.
Mathematicians handle this with zero new ideas. Slice the landscape in the x direction: you get an ordinary curve, take its ordinary slope. Repeat for y. These are called partial derivatives, because each one only does part of the job. That is the actual reason for the name.
drag the landscape to spin it. it likes that.
what just happened
Congratulations, you have now met multivariable calculus at a party. You do not have to marry it.
But keep this: when someone says gradient descent, they mean a ball rolling downhill on a landscape exactly like this one, and that is how neural networks learn. The next chapter removes the metaphor and hands you the ball.
New game+ · Gradient descent · ~4 min
A neural network is a ball rolling downhill. You already know the physics.
Chapter 6 left you a promise: the algorithm that trains neural networks is a ball rolling downhill. Here it is with the metaphor removed. A network's error is a landscape: every possible setting of its knobs has an error height, and training means finding the lowest valley.
The entire algorithm, no exaggeration: measure the slope where you stand (Chapter 1). Step downhill, sized by the slope. Repeat until flat. "Backpropagation" is the bookkeeping that computes the slope. Gradient descent is the step. A trillion-dollar industry runs on two lines you already understand.
The learning rate sets how big your steps are relative to the slope. Too small: training outlives you. Too big: the ball achieves escape velocity. A respectable fraction of machine learning practice is massaging this one number and calling it engineering.
Predict first
Steps are sized by the slope. So as the ball nears the bottom of a valley, its steps automatically get…
drag anywhere on the landscape to drop the ball. then let it roll.
what just happened
Three things your hands just learned, which people usually meet in year two of a degree: steps shrink near the bottom because the slope shrinks, so the ball parks itself. A too-large learning rate overshoots and explodes. And the ball only finds the valley it was dropped above: where you start matters, and the field has been haunted by that fact since day one.
Real networks have billions of knobs instead of one, so the landscape lives in a billion dimensions and the slope is a billion-entry list called the gradient. The ball does not care. Same rule: measure slope, step downhill, repeat. You now understand the training loop of every model you have ever talked to, including the one that built this page.
The cheat sheet · the entire course in six lines
Screenshot this. It's the whole trench coat.
- The limit. Shrink the window, watch the trend, name its destination. The only new idea.
- The derivative. Slope, right now. A speedometer. Written dy/dx.
- Zoom. Smooth curves are locally straight. That is why slopes exist at all.
- The integral. A total built from tiny pieces. An odometer. Written ∫.
- The Fundamental Theorem. Derivative and integral are inverses. The area grows at the rate of the height.
- Everything else is shortcut rules and costume jewelry.
– graded by the red pen, passed with distinction