Here’s my take on derivatives:

• We have a system to analyze, our function f
• The derivative f’ (df/dx) is the moment-by-moment behavior
• It turns out f is part of a bigger system (h = f + g)
• Using the behavior of the parts, can we figure out the behavior of the whole?

Yes. Every part has a “point of view” about how much change it added. Combine every point of view to get the overall behavior. Each derivative rule is an example of merging various points of view.

And why don’t we analyze the entire system at once? For the same reason you don’t eat a hamburger in one bite: small parts are easier to wrap your head around.

Instead of memorizing separate rules, let’s see how they fit together:

The goal is to really grok the notion of “combining perspectives”. This installment covers addition, multiplication, powers and the chain rule. Onward!

Functions: Anything, Anything But Graphs

The default calculus explanation writes “f(x) = x^2″ and shoves a graph in your face. Does this really help our intuition?

Not for me. Graphs squash input and output into a single curve, and hide the machinery that turns one into the other. But the derivative rules are about the machinery, so let’s see it!

I visualize a function as the process “input(x) => f => output(y)”.

It’s not just me. Check out this incredible, mechanical targetting computer (beginning of youtube series).

The machine computes functions like addition and multiplication with gears — you can see the mechanics unfolding!

Think of function f as a machine with an input lever “x” and an output lever “y”. As we adjust x, f sets the height for y. Another analogy: x is the input signal, f receives it, does some magic, and spits out signal y. Use whatever analogy helps it click.

Wiggle Wiggle Wiggle

The derivative is the “moment-by-moment” behavior of the function. What does that mean? (And don’t mindlessly mumble “The derivative is the slope”. See any graphs around these parts, fella?)

The derivative is how much we wiggle. The lever is at x, we “wiggle” it, and see how y changes. “Oh, we moved the input lever 1mm, and the output moved 5mm. Interesting.”

The result can be written “output wiggle per input wiggle” or “dy/dx” (5mm / 1mm = 5, in our case). This is usually a formula, not a static value, because it can depend on your current input setting.

For example, when f(x) = x^2, the derivative is 2x. Yep, you’ve memorized that. What does it mean?

If our input lever is at x = 10 and we wiggle it slightly (moving it by dx=0.1 to 10.1), the output should change by dy. How much, exactly?

• We know f’(x) = dy/dx = 2 * x
• At x = 10 the “output wiggle per input wiggle” is = 2 * 10 = 20. The output moves 20 units for every unit of input movement.
• If dx = 0.1, then dy = 20 * dx = 20 * .1 = 2

And indeed, the difference between 10^2 and (10.1)^2 is about 2. The derivative estimated how far the output lever would move (a perfect, infinitely small wiggle would move 2 units; we moved 2.01).

The key to understanding the derivative rules:

• Set up your system
• Wiggle each part of the system separately, see how far the output moves
• Combine the results

The total wiggle is the sum of wiggles from each part.

Addition and Subtraction

Time for our first system:

What happens when the input (x) changes?

In my head, I think “Function h takes a single input. It feeds the same input to f and g and adds the output levers. f and g wiggle independently, and don’t even know about each other!”

Function f knows it will contribute some wiggle (df), g knows it will contribute some wiggle (dg), and we, the prowling overseers that we are, know their individual moment-by-moment behaviors are added:

Again, let’s describe each “point of view”:

• The overall system has behavior dh
• From f’s perspective, it contributes df to the whole [it doesn't know about g]
• From g’s perspective, it contributes dg to the whole [it doesn't know about f]

Every change to a system is due to some part changing (f and g). If we add the contributions from each possible variable, we’ve described the entire system.

df vs df/dx

Sometimes we use df, other times df/dx — what gives? (This confused me for a while)

• df is a general notion of “however much f changed”
• df/dx is a specific notion of “however much f changed, in terms of how much x changed”

The generic “df” helps us see the overall behavior.

An analogy: Imagine you’re driving cross-country and want to measure the fuel efficiency of your car. You’d measure the distance traveled, check your tank to see how much gas you used, and finally do the division to compute “miles per gallon”. You measured distance and gasoline separately — you didn’t jump into the gas tank to get the rate on the go!

In calculus, sometimes we want to think about the actual change, not the ratio. Working at the “df” level gives us room to think about how the function wiggles overall. We can eventually scale it down in terms of a specific input.

And we’ll do that now. The addition rule above can be written, on a “per dx” basis, as:

Multiplication (Product Rule)

Next puzzle: suppose our system multiplies parts “f” and g”. How does it behave?

Hrm, tricky — the parts are interacting more closely. But the strategy is the same: see how each part contributes from its own point of view, and combine them:

• total change in h = f’s contribution (from f’s point of view) + g’s contribution (from g’s point of view)

Check out this diagram:

What’s going on?

• We have our system: f and g are multiplied, giving h (the area of the rectangle)
• Input “x” changes by dx off in the distance. f changes by some amount df (think absolute change, not the rate!). Similarly, g changes by its own amount dg. Because f and g changed, the area of the rectangle changes too.
• What’s the area change from f’s point of view? Well, f knows he changed by df, but has no idea what happened to g. From f’s perspective, he’s the only one who moved and will add a slice of area = df * g
• Similarly, g doesn’t know how f changed, but knows he’ll add as slice of area “dg * f”

The overall change in the system (dh) is the two slices of area:

Now, like our miles per gallon example, we “divide by dx” to write this in terms of how much x changed:

(Aside: Divide by dx? Engineers will nod, mathematicians will frown. Technically, df/dx is not a fraction: it’s the entire operation of taking the derivative (with the limit and all that). But infinitesimal-wise, intuition-wise, we are “scaling by dx”. I’m a smiler.)

The key to the product rule: add two “slivers of area”, one from each point of view.

Gotcha: But isn’t there some effect from both f and g changing simultaneously (df * dg)?

Yep. However, this area is an infinitesimal * infinitesimal (a “2nd-order infinitesimal”) and invisible at the current level. It’s a tricky concept, but (df * dg) / dx vanishes compared to normal derivatives like df/dx. We vary f and g indepdendently and combine the results, and ignore results from them moving together.

The Chain Rule: It’s Not So Bad

Let’s say g depends on f, which depends on x:

The chain rule lets us “zoom into” a function and see how an initial change (x) can effect the final result down the line (g).

Interpretation 1: Convert the rates

A common interpretation is to multiply the rates:

x wiggles f. This creates a rate of change of df/dx, which wiggles g by dg/df. The entire wiggle is then:

This is similar to the “factor-label” method in chemistry class:

If your “miles per second” rate changes, multiply by the conversion factor to get the new “miles per hour”. The second doesn’t know about the hour directly — it goes through the second => minute conversion.

Similarly, g doesn’t know about x directly, only f. Function g knows it should scale its input by dg/df to get the output. The initial rate (df/dx) gets modified as it moves up the chain.

Interpretation 2: Convert the wiggle

I prefer to see the chain rule on the “per-wiggle” basis:

• x wiggles by dx, so
• f wiggles by df, so
• g wiggles by dg

Cool. But how are they actually related? Oh yeah, the derivative! (It’s the output wiggle per input wiggle):

Remember, the derivative of f (df/dx) is how much to scale the initial wiggle. And the same happens to g:

It will scale whatever wiggle comes along its input lever (f) by dg/df. If we write the df wiggle in terms of dx:

We have another version of the chain rule: dx starts the chain, which results in some final result dg. If we want the final wiggle in terms of dx, divide both sides by dx:

The chain rule isn’t just factor-label unit cancellation — it’s the propagation of a wiggle, which gets adjusted at each step.

The chain rule works for several variables (a depends on b depends on c), just propagate the wiggle as you.

Try to imagine “zooming into” different variable’s point of view. Starting from dx and looking up, you see the entire chain of transformations needed before the impulse reaches g.

Chain Rule: Example Time

Let’s say we put a “squaring machine” in front of a “cubing machine”:

input(x) => f:x^2 => g:f^3 => output(y)

f:x^2 means f squares its input. g:f^3 means g cubes its input, the value of f. For example:

input(2) => f(2) => g(4) => output:64

Start with 2, f squares it (2^2 = 4), and g cubes this (4^3 = 64). It’s a 6th power machine:

And what’s the derivative?

• f changes its input wiggle by df/dx = 2x
• g changes its input wiggle by dg/df = 3f^2

The final change is:

Chain Rule: Gotchas

Functions treat their inputs like a blob

In the example, g’s derivative (“x^3 = 3x^2″) doesn’t refer to the original “x”, just whatever the input was (foo^3 = 3*foo^2). The input was f, and it treats f as a single value. Later on, we scurry in and rewrite f in terms of x. But g has no involvement with that — it doesn’t care that f can be rewritten in terms of smaller pieces.

In many examples, the variable “x” is the “end of the line”.

Questions ask for df/dx, i.e. “Give me changes from x’s point of view”. Now, x could depend on something deeper variable, but that’s not being asked for. It’s like saying “I want miles per hour. I don’t care about miles per minute or miles per second. Just give me miles per hour”. df/dx means “stop looking at inputs once you get to x”.

How come we multiply derivatives with the chain rule, but add them for the others?

The regular rules are about combining points of view to get an overall picture. What change does f see? What change does g see? Add them up for the total.

The chain rule is about going deeper into a single part (like f) and seeing if it’s controlled by another variable. It’s like looking inside a clock and saying “Hey, the minute hand is controlled by the second hand!”. We’re staying inside the same part.

Sure, eventually this “per-second” perspective of f could be added to some perspective from g. Great. But the chain rule is about diving deeper into “f’s” root causes.

Power Rule: Oft Memorized, Seldom Understood

What’s the derivative of x^4? 4x^3? Great. You brought down the exponent and subtracted one. Now explain why!

Hrm. There’s a few approaches, but here’s my new favorite: x^4 is really x * x * x * x. It’s the multiplication of 4 “independent” variables. Each x doesn’t know about the others, it might as well be x * u * v * w.

Now think about the first x’s point of view:

• It changes from x to x + dx
• The change in the overall function is [(x + dx) - x][u * v * w] = dx[u * v * w]
• The change on a “per dx” basis is [u * v * w]

Similarly,

• From u’s point of view, it changes by du. It contributes (du/dx)*[x * v * w] on a “per dx” basis
• v contributes (dv/dx) * [x * u * w]
• w contributes (dw/dx) * [x * u * v]

The curtain is unveiled: x, u, v, and w are the same! The “point of view” conversion factor is 1 (du/dx = dv/dx = dw/dx = dx/dx = 1), and the total change is

In a sentence: the derivative of x^4 is 4x^3 because x^4 has four identical “points of view” which are being combined. Booyeah!

Take A Breather

I hope you’re seeing the derivative in a new light: we have a system of parts, we wiggle our input and see how the whole thing moves. It’s about combining perspectives: what does each part add to the whole?

In the follow-up article, we’ll look at even more powerful rules (exponents, quotients, and friends). Happy math.

Psst! The derivative is the heart of calculus, buried inside this definition:

But what does it mean?

Let's say I gave you a magic newspaper that listed the daily stock market changes for the next few years (+1% Monday, -2% Tuesday...). What could you do?

Well, you'd apply the changes one-by-one, plot out future prices, and buy low / sell high to build your empire. You could even hire away the monkeys who currently throw darts at newspapers.

Others call the derivative "the slope of a function" -- it's so bland! Like the stock list, the derivative is a total, predictive understanding of a system. You can plot the past/present/future, find minimums/maximums, and yes, staff your simian workforce.

Step away from the gnarly equation. Equations exist to convey ideas: understand the idea, not the grammar.

Derivatives create a perfect model of change from an imperfect guess.

This result came over thousands of years of thinking, from Archimedes to Newton. Let's look at the analogies behind it.

We all live in a shiny continuum

Infinity is a constant source of paradoxes ("headaches"):

• A line is made up of points? Sure.
• So there's an infinite number of points on a line? Yep.
• How do you cross a room when there's an infinite number of points to visit? (Gee, thanks Zeno).

And yet, we move. My intuition is to fight infinity with infinity. Sure, there's infinity points between 0 and 1. But I move two infinities of points per second (somehow!) and I cross the gap in half a second.

Distance has infinite points, motion is possible, therefore motion is in terms of "infinities of points per second".

Instead of thinking of differences ("How far to the next point?") we can compare rates ("How fast are you moving through this continuum?").

It's strange, but you can see 10/5 as "I need to travel 10 'infinities' in 5 segments of time. To do this, I travel 2 'infinities' for each unit of time".

Analogy: See division as a rate of motion through a continuum of points

What's after zero?

Another brain-buster: What number comes after zero? .01? .0001?

Hrm. Anything you can name, I can name smaller (I'll just halve your number... nyah!).

Even though we can't calculate the number after zero, it must be there, right? Like demons of yore, it's the "number that cannot be written, lest ye be smitten".

Call the gap to the next number "dx". I don't know exactly how big it is, but it's there!

Analogy: dx is a "jump" to the next number in the continuum.

Measurements depend on the instrument

The derivative predicts change. Ok, how do we measure speed (change in distance)?

Officer: Do you know how fast you were going?

Driver: I have no idea.

Officer: 95 miles per hour.

Driver: But I haven't been driving for an hour!

We clearly don't need a "full hour" to measure your speed. We can take a before-and-after measurement (over 1 second, let's say) and get your instantaneous speed. If you moved 140 feet in one second, you're going ~95mph. Simple, right?

Not exactly. Imagine a video camera pointed at Clark Kent (Superman's alter-ego). The camera records 24 pictures/sec (40ms per photo) and Clark seems still. On a second-by-second basis, he's not moving, and his speed is 0mph.

Wrong again! Between each photo, within that 40ms, Clark changes to Superman, solves crimes, and returns to his chair for a nice photo. We measured 0mph but he's really moving -- he goes too fast for our instruments!

Analogy: Like a camera watching Superman, the speed we measure depends on the instrument!

Running the Treadmill

We're nearing the chewy, slightly tangy center of the derivative. We need before-and-after measurements to detect change, but our measurements could be flawed.

Imagine a shirtless Santa on a treadmill (go on, I'll wait). We're going to measure his heart rate in a stress test: we attach dozens of heavy, cold electrodes and get him jogging.

Santa huffs, he puffs, and his heart rate shoots to 190 beats per minute. That must be his "under stress" heart rate, correct?

Nope. See, the very presence of stern scientists and cold electrodes increased his heart rate! We measured 190bpm, but who knows what we'd see if the electrodes weren't there! Of course, if the electrodes weren't there, we wouldn't have a measurement.

What to do? Well, look at the system:

• measurement = actual amount + measurement effect

Ah. After lots of studies, we may find "Oh, each electrode adds 10bpm to the heartrate". We make the measurement (imperfect guess of 190) and remove the effect of electrodes ("perfect estimate").

Analogy: Remove the "electrode effect" after making your measurement

By the way, the "electrode effect" shows up everywhere. Research studies have the Hawthorne Effect where people change their behavior because they are being studied. Gee, it seems everyone we scrutinize sticks to their diet!

Understanding the derivative

Armed with these insights, we can see how the derivative models change:

Start with some system to study, f(x):

1. Change by the smallest amount possible (dx)
2. Get the before-and-after difference: f(x + dx) - f(x)
3. We don't know exactly how small "dx" is, and we don't care: get the rate of motion through the continuum: [f(x + dx) - f(x)] / dx
4. This rate, however small, has some error (our cameras are too slow!). Predict what happens if the measurement were perfect, if dx wasn't there.

The magic's in the final step: how do we remove the electrodes? We have two approaches:

• Limits: what happens when dx shrinks to nothingness, beyond any error margin?
• Infinitesimals: What if dx is a tiny number, undetectable in our number system?

Both are ways to formalize the notion of "How do we throw away dx when it's not needed?".

My pet peeve: Limits are a modern formalism, they didn't exist in Newton's time. They help make dx disappear "cleanly". But teaching them before the derivative is like showing a steering wheel without a car! It's a tool to help the derivative work, not something to be studied in a vacuum.

An Example: f(x) = x^2

Let's shake loose the cobwebs with an example. How does the function f(x) = x^2 change as we move through the continuum?

Note the difference in the last 2 equations:

• One has the error built in (dx)
• The other has the "true" change, where dx = 0 (our measurements have no effect on the outcome)

Time for real numbers. Here's the values for f(x) = x^2, with intervals of dx = 1:

• 1, 4, 9, 16, 25, 36, 49, 64...

The absolute change between each result is:

• 1, 3, 5, 7, 9, 11, 13, 15...

(Here, the absolute change is the "speed" between each step, where the interval is 1)

Consider the jump from x=2 to x=3 (3^2 - 2^2 = 5). What is "5" made of?

• Measured rate = Actual Rate + Error
• 5 = 2x + dx
• 5 = 2(2) + 1

Sure, we measured a "5 units moved per second" because we went from 4 to 9 in one interval. But our instruments trick us! 4 units of speed came from the real change, and 1 unit was due to shoddy instruments (1.0 is a large jump, no?).

If we restrict ourselves to integers, 5 is the perfect speed measurement from 4 to 9. There's no "error" in assuming dx = 1 because that's the true interval between neighboring points.

But in the real world, measurements every 1.0 seconds is too slow. What if our dx was 0.1? What speed would we measure at x=2?

Well, we examine the change from x=2 to x=2.1:

• 2.1^2 - 2^2 = 0.41

Remember, 0.41 is what we changed in an interval of 0.1. Our speed-per-unit is 0.41 / .1 = 4.1. And again we have:

• Measured rate = Actual Rate + Error
• 4.1 = 2x + dx

Interesting. With dx=0.1, the measured and actual rates are close (4.1 to 4, 2.5% error). When dx=1, the rates are pretty different (5 to 4, 25% error).

Following the pattern, we see that throwing out the electrodes (letting dx=0) reveals the true rate of 2x.

In plain English: We analyzed how f(x) = x^2 changes, found an "imperfect" measurement of 2x + dx, and deduced a "perfect" model of change as 2x.

The derivative as "continuous division"

I see the integral as better multiplication, where you can apply a changing quantity to another.

The derivative is "better division", where you get the speed through the continuum at every instant. Something like 10/5 = 2 says "you have a constant speed of 2 through the continuum".

When your speed changes as you go, you need to describe your speed at each instant. That's the derivative.

If you apply this changing speed to each instant (take the integral of the derivative), you recreate the original behavior, just like applying the daily stock market changes to recreate the full price history. But this is a big topic for another day.

Gotcha: The Many meanings of "Derivative"

You'll see "derivative" in many contexts:

• "The derivative of x^2 is 2x" means "At every point, we are changing by a speed of 2x (twice the current x-position)". (General formula for change)

• "The derivative is 44" means "At our current location, our rate of change is 44." When f(x) = x^2, at x=22 we're changing at 44 (Specific rate of change).

• "The derivative is dx" may refer to the tiny, hypothetical jump to the next position. Technically, dx is the "differential" but the terms get mixed up. Sometimes people will say "derivative of x" and mean dx.

Gotcha: Our models may not be perfect

We found the "perfect" model by making a measurement and improving it. Sometimes, this isn't good enough -- we're predicting what would happen if dx wasn't there, but added dx to get our initial guess!

Some ill-behaved functions defy the prediction: there's a difference between removing dx with the limit and what actually happens at that instant. These are called "discontinuous" functions, which is essentially "cannot be modeled with limits". As you can guess, the derivative doesn't work on them because we can't actually predict their behavior.

Discontinuous functions are rare in practice, and often exist as "Gotcha!" test questions ("Oh, you tried to take the derivative of a discontinuous function, you fail"). Realize the theoretical limitation of derivatives, and then realize their practical use in measuring every natural phenomena. Nearly every function you'll see (sine, cosine, e, polynomials, etc.) is continuous.

Gotcha: Integration doesn't really exist

The relationship between derivatives, integrals and anti-derivatives is nuanced (and I got it wrong originally). Here's a metaphor. Start with a plate, your function to examine:

• Differentiation is breaking the plate into shards. There is a specific procedure: take a difference, find a rate of change, then assume dx isn't there.
• Integration is weighing the shards: your original function was "this" big. There's a procedure, cumulative addition, but it doesn't tell you what the plate looked like.
• Anti-differentiation is figuring out the original shape of the plate from the pile of shards.

There's no algorithm to find the anti-derivative; we have to guess. We make a lookup table with a bunch of known derivatives (original plate => pile of shards) and look at our existing pile to see if it's similar. "Let's find the integral of 10x. Well, it looks like 2x is the derivative of x^2. So... scribble scribble... 10x is the derivative of 5x^2.".

Finding derivatives is mechanics; finding anti-derivatives is an art. Sometimes we get stuck: we take the changes, apply them piece by piece, and mechanically reconstruct a pattern. It might not be the "real" original plate, but is good enough to work with.

Another subtlety: aren't the integral and anti-derivative the same? (That's what I originally thought)

Yes, but this isn't obvious: it's the fundamental theorem of calculus! (It's like saying "Aren't a^2 + b^2 and c^2 the same? Yes, but this isn't obvious: it's the Pythagorean theorem!"). Thanks to Joshua Zucker for helping sort me out.

Reading math

Math is a language, and I want to "read" calculus (not "recite" calculus, i.e. like we can recite medieval German hymns). I need the message behind the definitions.

My biggest aha! was realizing the transient role of dx: it makes a measurement, and is removed to make a perfect model. Limits/infinitesimals are a formalism, we can't get caught up in them. Newton seemed to do ok without them.

Armed with these analogies, other math questions become interesting:

• How do we measure different sizes of infinity? (In some sense they're all "infinite", in other senses the range (0,1) is smaller than (0,2))
• What are the real rules about making "dx go away"? (How do infinitesimals and limits really work?)
• How do we describe numbers without writing them down? "The next number after 0" is the beginnings of analysis (which I want to learn).

The fundamentals are interesting when you see why they exist. Happy math.

I was stuck thinking sine had to be extracted from other shapes. A quick analogy:

You: Geometry is about shapes, lines, and so on...

Alien: Oh? Can you show me a line?

You (looking around): Uh... see that brick, there? A line is one edge of that brick.

Alien: So lines are part of a shape?

You: Sort of. Yes, most shapes have lines in them. But a line is a basic concept on its own: a beam of light, a direct route on a map, or even a vector in many dimensions. You see--

Alien: Lines come from bricks. Bricks bricks bricks.

The frustration! Because sine is introduced with angles and circles, my brain thinks "Sine comes from circles. Circles circles circles."

No more. In a sentence:

Sine a natural sway, the epitome of smoothness: it makes circles "circular" in the same way lines make squares "square".

Let's build our intuition by seeing sine as its own shape, and then understand how it fits into circles and the like. Onward!

Sine vs Lines

Remember to separate an idea from an example: squares are examples of lines. Sine clicked when it became its own idea, not "part of a circle."

Let's observe sine in a simulator (Email readers, you may need to open the article directly):

Hubert will give the tour:

• Click start. Go, Hubert go! Notice that smooth back and forth motion? That's Hubert, but more importantly (sorry Hubert), that's sine! It's natural, the way springs bounce, pendulums swing, strings vibrate... and many things move.
• Change "vertical" to "linear". Big difference -- see how the motion gets constant and robotic, like a game of pong?

Let's explore the differences with video:

• Linear motion is constant: we go a set speed and turn around instantly. It's the unnatural motion in the robot dance (notice the linear bounce with no slowdown at 0:07, the strobing effect at :38).

• Sine changes its speed: it starts fast, slows down, stops, and speeds up again. It's the enchanting smoothness in liquid dancing (human sine wave at 0:12 and 0:23, natural bounce at :47).

Unfortunately, textbooks don't show sine with animations or dancing. No, they prefer to introduce sine with a timeline (try setting "horizontal" to "timeline"):

(source)

Egads. This is the schematic diagram we've always been shown. Does it give you the feeling of sine? Not any more than a skeleton portrays the agility of a cat. Let's watch sine move and then chart its course.

The Unavoidable Circle

Circles have sine. Yes. But seeing the sine inside a circle is like getting the eggs back out of the omelette. It's all mixed together!

Let's take it slow. In the simulation, set Hubert to vertical:none and horizontal: sine*. See him wiggle sideways? That's the motion of sine. There's a small tweak: normally sine starts the cycle at the neutral midpoint and races to the max. This time, we start at the max and fall towards the midpoint. Sine that "starts at the max" is called cosine, and it's just a version of sine (like a horizontal line is a version of a vertical line).

Ok. Time for both sine waves: put vertical as "sine" and horizontal as "sine*". And... we have a circle!

A horizontal and vertical "spring" combine to give circular motion. Most textbooks draw the circle and try to extract the sine, but I prefer to build up: start with pure horizontal or vertical motion and add in the other.

Quick Q & A

A few insights I missed when first learning sine:

Sine really is 1-dimensional

Sine wiggles in one dimension. Really. We often graph sine over time (so we don't write over ourselves) and sometimes the "thing" doing sine is also moving, but this is optional! A spring in one dimension is a perfectly happy sine wave.

Circles are an example of two sine waves

Circles and square are a combination of basic components (sines and lines). But circles aren't the "origin" of sines any more than squares are the root cause of lines.

What do the values of sine mean?

Sine cycles between -1 and 1. It starts at 0, grows to 1.0 (max), dives to -1.0 (min) and returns to neutral. I also see sine like a percentage, from 100% (full steam ahead) to -100% (full retreat).

What's is the input 'x' in sin(x)?

Tricky question. Sine is a cycle and x, the input, is how far along we are in the cycle.

Let's look at lines:

• You're traveling on a square. Each side takes 10 seconds.
• After 1 second, you are 10% complete on that side
• After 5 seconds, you are 50% complete
• After 10 seconds, you finished the side

Linear motion has few surprises. Now for sine (focusing on the "0 to max" cycle):

• We're traveling on a sine wave, from 0 (neutral) to 1.0 (max). This portion takes 10 seconds.
• After 5 seconds we are... 70% complete! Sine rockets out of the gate and slows down. Most of the gains are in the first 5 seconds
• It takes 5 more seconds to get from 70% to 100%. And going from 98% to 100% takes almost a full second!

Despite our initial speed, sine slows so we gently kiss the max value before turning around. This smoothness makes sine, sine.

For the geeks: Press "show stats" in the simulation. You'll see the percent complete of the total cycle, mini-cycle (0 to 1.0), and the value attained so far. Stop, step through, and switch between linear and sine motion to see the values.

Quick quiz: What's higher, 10% of a linear cycle, or 10% of a sine cycle? Sine. Remember, it barrels out of the gate at max speed. The average speed is indeed hit at 50% of the cycle time, but in the beginning we're going faster than average.

So x is the 'amount of your cycle'. What's the cycle?

It depends on the context.

• Basic trig: 'x' is degrees, and a full cycle is 360 degrees
• Advanced trig: 'x' is radians (they are more natural!), and a full cycle is going around the unit circle (2*pi radians)

Play with values of x here:

But again, cycles depend on circles! Can we escape their tyranny?

Pi without Pictures

Imagine a sightless alien who only notices shades of light and dark. Could you describe pi to him? It's hard to flicker the idea of a circle's circumference, right?

Let's step back a bit. Sine is a repeating pattern, which means it must... repeat! It goes from 0, to 1, to 0, to -1, to 0, and so on.

Let's define pi as the time sine takes from 0 to 1 and back to 0. Whoa! Now we're using pi without a circle too! Pi is a concept that just happens to show up in circles:

• Sine is a gentle back and forth rocking
• Pi is the time from neutral to max and back to neutral
• n*Pi (0*Pi, 1*pi, 2*pi, and so on) are the times you are at neutral
• 2*Pi, 4*pi, 6*pi, etc. are full cycles

Aha! That is why pi appears in so many formulas! Pi doesn't "belong" to circles any more than 0 and 1 do -- pi is about sine returning to center! A circle is an example of a shape that repeats and returns to center every 2*pi units. But springs, vibrations, etc. return to center after pi too!

Question: If pi is half of a natural cycle, why does it go on forever (i.e., irrational)?

Can I answer a question with a question? Why does the diagonal of a "unit square" have length sqrt(2), which also goes on forever?

But yes, I realize it's philosophically inconvenient when nature behaves randomly. I don't have a good intuition.

How fast is sine?

I've been tricky. Previously, I said "imagine it takes sine 10 seconds from 0 to max". And now it's pi seconds from 0 to max back to 0? What gives?

• sin(x) is the default, off-the-shelf sine wave, that indeed takes pi units of time from 0 to max to 0 (or 2*pi for a complete cycle)
• sin(2x) is a wave that moves twice as fast
• sin(x/2) is a wave that moves twice as slow

So, we use sin(n*x) to get a sine wave cycling as fast as we need. Often, the phrase "sine wave" is referencing the general shape and not a specific speed.

Part 2: Understanding the definitions of sine

That's a brainful -- take a break if you need it. Hopefully, sine is emerging as its own pattern. Now let's develop our intuition by seeing how common definitions of sine connect.

Definition 1: The height of a triangle / circle!

Sine was first found in triangles. You may remember "SOH CAH TOA" as a mnemonic

• SOH: Sine is Opposite / Hypotenuse
• CAH: Cosine is Adjacent / Hypotenuse
• TOA: Tangent is Opposite / Adjacent

For a right triangle with angle x, sin(x) is the length of the opposite side divided by the hypotenuse. If we make the hypotenuse 1, we can simplify to:

• Sine = Opposite
• Cosine = Adjacent

And with more cleverness, we can draw our triangles with hypotenuse 1 in a circle with radius 1:

Voila! A circle containing all possible right triangles (since they can be scaled up using similarity). For example:

• sin(45) = .707
• Lay down a 10-foot pole and raise it 45 degrees. It is 10 * sin(45) = 7.07 feet off the ground
• An 8-foot pole would be 8 * sin(45) = 5.65 feet

These direct manipulations are great for construction (the pyramids won't calculate themselves). Unfortunately, after thousands of years we start thinking the meaning of sine is the height of a triangle. No no, it's a shape that shows up in circles (and triangles).

Realistically, for many problems we go into "geometry mode" and start thinking "sine = height" to speed through things. That's fine -- just don't get stuck there.

Definition 2: The infinite series

I've avoided the elephant in the room: how in blazes do we actually calculate sine!? Is my calculator drawing a circle and measuring it?

Glad to rile you up. Here's the circle-less secret of sine:

Sine is acceleration opposite to your current position

Using our bank account metaphor: Imagine a perverse boss who gives you a raise the exact opposite of your current bank account! If you have $50 in the bank, then your raise next week is -$50. Of course, your income might be $75/week, so you'll still be earning some money ($75 - $50 for that week), but eventually your balance will decrease as the "raises" overpower your income.

But never fear! Once your account hits negative (say you're at -$50), then your boss gives a legit $50/week raise. Again, your income might be negative, but eventually the raises will overpower it.

This constant pull towards the center keeps the cycle going: when you rise up, the "pull" conspires to pull you in again. It also explains why neutral is the max speed for sine: If you are at the max, you begin falling and accumulating more and more "negative raises" as you plummet. As you pass through then neutral point you are feeling all the negative raises possible (once you cross, you'll start getting positive raises and slowing down).

By the way: since sine is acceleration opposite to your current position, and a circle is made up of a horizontal and vertical sine... you got it! Circular motion can be described as "a constant pull opposite your current position, towards your horizontal and vertical center".

Geeking Out With Calculus

Let's describe sine with calculus. Like e, we can break sine into smaller effects:

• Start at 0 and grow at unit speed
• At every instant, get pulled back by negative acceleration

How should we think about this? See how each effect above changes our distance from center:

• Our initial kick increases distance linearly: y (distance from center) = x (time taken)
• At any moment, we feel a restoring force of -x. We integrate twice to turn negative acceleration into distance:

Seeing how acceleration impacts distance is like seeing how a raise hits your bank account. The "raise" must change your income, and your income changes your bank account (two integrals "up the chain").

So, after "x" seconds we might guess that sine is "x" (initial impulse) minus x^3/3! (effect of the acceleration):

Something's wrong -- sine doesn't nosedive! With e, we saw that "interest earns interest" and sine is similar. The "restoring force" changes our distance by -x^3/3!, which creates another restoring force to consider. Consider a spring: the pull that yanks you down goes too far, which shoots you downward and creates another pull to bring you up (which again goes too far). Springs are crazy!

We need to consider every restoring force:

• y = x is our initial motion, which creates a restoring force of impact:
• y = -x^3/3!, which creates a restoring force of impact:
• y = x^5/5!, which creates a restoring force of impact:
• y = -x^7/7! which creates a restoring force of impact...

Just like e, sine can be described with an infinite equation:

I saw this formula a lot, but it only clicked when I saw sine as a combination of an initial impulse and restoring forces. The initial push (y = x, going positive) is eventually overcome by a restoring force (which pulls us negative), which is overpowered by its own restoring force (which pulls us positive), and so on.

A few fun notes:

• Consider the "restoring force" like "positive or negative interest". This makes the sine/e connection in Euler's formula easier to understand. Sine is like e, except sometimes it earns negative interest. There's more to learn here .
• For small amounts, "y = x" is a good guess for sine. We just take the initial impulse and ignore any restoring forces.

The Calculus of Cosine

Cosine is just a shifted sine, and is fun (yes!) now that we understand sine:

• Sine: Start at 0, initial impulse of y = x (100%)
• Cosine: Start at 1, no initial impulse

So cosine just starts off... sitting there at 1. We let the restoring force do the work:

Again, we integrate -1 twice to get -x^2/2!. But this kicks off another restoring force, and before you know it:

Definition 3: The differential equation

We've described sine's behavior with specific equations. A more succinct way (equation):

This beauty says:

• Our current position is y
• Our acceleration (2nd derivative, or y'') is the opposite of our current position (-y)

Both sine and cosine make this true. I first hated this definition; it's so divorced from a visualization. I didn't realize it described the essence of sine, "acceleration opposite your position".

And remember how sine and e are connected? Well, e^x can be be described by (equation):

The same equation with a positive sign ("acceleration equal to your position")! When sine is "the height of a circle" it's really hard to make the connection to e.

One of my great mathematical regrets is not learning differential equations. But I want to, and I suspect having an intuition for sine and e will be crucial.

Summing it up

The goal is to move sine from some mathematical trivia ("part of a circle") to its own shape:

• Sine is a smooth, swaying motion between min (-1) and max (1). Mathematically, you're accelerating opposite your position. This "negative interest" keeps sine rocking forever.
• Sine happens to appear in circles and triangles (and springs, pendulums, vibrations, sound...)
• Pi is the time from neutral to neutral in sin(x). It doesn't "belong" to circles any more than 0 and 1 do.

Let sine enter your mental toolbox (Hrm, I need a formula to make smooth changes...). Eventually, we'll understand the foundations intuitively (e, pi, radians, imaginaries, sine...) and they can be mixed into a scrumptious math salad. Enjoy!

I love physics, but it's not the best lead-in. It makes us wait till science class (9th grade?) and worse, it implies calculus is "math for science class". Couldn't we introduce the themes to 5th graders, and relate it to everyday life?

I think so. So here's the goal:

• Use money, not physics, to introduce calculus concepts
• Explore how patterns relate (bank account to salary; salary to raises)
• Use our intuition to explore potential issues (can we keep drilling into patterns?)

Strap on your math helmet, time to dive in.

Money money money

My favorite calculus example is the relationship between your bank account, salary, and raises.

Here's Joe ("Hi, Joe"). You, the sly scoundrel you are, sneak onto Joe's computer and monitor his bank account each week. What can you learn?

Ack. Clearly, not much happened -- Joe isn't earning anything. And what if you see this?

Easy enough: Joe's making some money. And how much? With a quick subtraction, we can figure out his weekly paycheck. Turns out Joe is making a steady $100/week.

• Key idea: If I know your bank account, I know your salary

The bank account is dependent on the salary -- it changes because of the weekly salary.

Raise the roof

Let's go deeper: knowing the salary, what else can we figure out? Well, the salary is another pattern to analyze -- we can see if it changes! That is, we can tell if Joe's salary is changing week by week (is he getting a raise?).

The process:

• Look at Joe's weekly bank account
• Take the difference in bank account to get the weekly salary
• Take the difference in salary to get the weekly raise (if any)

In the first example ($100/week), it's clear there's no raise (sorry, Joe). The main idea is to "take the difference" to analyze the first pattern (bank account to salary) and "take the difference again" to find yet another pattern (salary to raise).

Working backwards

We just went "down", from bank account to salary. Does it work the other way: knowing the salary, can I predict the bank account?

You're hesitating, I can tell. Yes, knowing Joe gets $100/week is nice. But... don't we need to know the starting account balance?

Yes! The changes to his account (salary) is not enough -- where did it start? For simplicity (i.e., what you see in homework problems) we often assume Joe starts with $0. But, if you are actually making a prediction, you want to know the initial conditions (the "+ C").

A More Complex Pattern

Let's say Joe's account grows like this: 100, 300, 600, 1000, 1500...

What's going on? Is it random? Well, we can do our week-by-week subtraction to get this:

Interesting -- Joe's income is changing each week. We do another week-by-week difference and get this:

And yep, Joe's getting a steady raise of $100/week. Let's get wild and chart them on the same graph:

One way to think about it: Joe gets a raise each week, which changes his salary, which changes his bank account. As the raises continue to appear, his salary continues to increase and his bank account rises. You can almost think of the raise "pushing up" the salary, which "pushes up" the bank account.

So... Where's the Calculus?

What's the formula for Joe's bank account for any week? Well, it's the sum of his salaries up to that point:

100 + 200 + 300 + 400... = 100 * n * (n + 1)/2

The formula for adding up a series of numbers (1 + 2 + 3 + 4...) is very close to n^2/2, and gets closer as the number of steps increases.

This is our first "calculus" relationship:

• A constant raise ($100/week) leads to a...
• Linear increase in salary (100, 200, 300, 400) which leads to a...
• Quadratic (something * n^2) increase in bank account (100, 300, 600, 1000... you see it curve!)

Now, why is it roughly 1/2 * n² and not n²? One intuition: The linear increase in salary (100, 200, 300) gives us a triangle. The area of the triangle represents all the payments so far, and the area is 1/2 * base * height. The base is n (the number of weeks) and the height (income) is 100 * n.

Geometric arguments get more difficult in higher dimensions -- just because we can work out 2*100 with addition doesn't mean it's the easiest way. Calculus gives us the rules to jump between patterns (taking derivatives and integrals).

Points to Explore

Our understanding of bank accounts, salaries, and raises lets us explore deeper.

Could we figure out the total earnings between weeks 1 and 10?

Sure! There's two ways: we could add up our income for each week (week 1 salary + week 2 salary + week 3 salary...) or just subtract the bank account (Week 10 bank account - week 1 bank account). This idea has a beefy name: the Fundamental Theorem of Calculus!

Can we keep going "down" (taking derivatives) beyond the raise?

Well, why not? If the raise is $100/week, if we take the difference again we see it drops to 0 (there is no "raise raise", aka the raise is always steady). But, we can imagine the case where the raise itself is raising (week1 raise = 100, week2 raise = 200). Using our intuition: if the "raise raise" is constant, the raise is linear (something * n), the income is quadratic (something * n²) and the bank account is cubic (something * n³). And yes, it's true!

Can derivatives go on forever?

Yep. Maybe the connection is bank account => salary => raise => inflation => milk output of Farmer Joe's cow => how much Joe feeds the cow each week. Many patterns "stop having derivatives" once we get to the root cause. But certain interesting patterns, like exponential growth, have an infinite number of components! You have interest, which earns interest, which earns interest, which earns interest... forever! You can never find the single "root cause" of your bank account because an infinite number of components went into it (pretty trippy).

What happens if the raise goes negative?

Interesting question. As the raise goes negative, his salary will start lowering. But, as long as the salary is above zero, the bank account will keep rising! After all, going from $200 to $100 per week, while bad to you, still helps your bank account. Eventually, a negative raise will overpower the salary, making it negative, which means Joe is now paying his employer. But up until that point, Joe's bank account would be growing.

How quickly can we check for differences?

Suppose we're measuring a stock portfolio, not a bank account. We might want a second-by-second model of our salary and account balance. The idea is to measure at intervals short enough to get the detail we need -- a large aspect of calculus is deciding what "limit" is enough to say "Ok, this is accurate enough for me!".

The calculus formulas you typically see (integral of x = 1/2 * x^2) are different from the "discrete" formulas (sum of 1 to n = 1/2 * n * (n + 1)) because the discrete case is using "chunky" intervals.

Key Takeaways

Why do I care about the analogy used? The traditional "distance, velocity, acceleration" doesn't lead to the right questions. What's the next derivative of acceleration? (It's called "jerk", and it's rarely used). Such a literal example is like having kids think multiplication is only for finding area, and only works on two numbers at a time.

Here's the key points:

• Calculus helps us find related patterns (bank account, to salary, to raises)
• The "derivative" is going "down" (finding week-by-week changes to get your salary)
• The "integral" is going "up" (adding up your salary to get your bank account)
• We can figure out a formula for a pattern (given my bank account, predict my salary) or get a specific value (what's my salary at week 3?)
• Calculus is useful outside the hard sciences. If you have a pattern or formula (production rate, size of a population, GDP of a country) and want to examine its behavior, calculus is the tool for you.
• Textbook calculus involves memorizing the rules to derive and integrate formulas. Learn the basics (x^n, e, ln, sin, cos) and leave the rest to machines. Our brainpower is better spent learning how to translate our thoughts into the language of math.

In my fantasy world, derivatives and integrals are just two everyday concepts. They're "what you can do" to formulas, just like addition and subtraction are "what you can do" to numbers.

"Hey kids, we find the total mass using addition (Mass1 + Mass2 = Mass3). And to find out how our position changes, we use the derivative".

"Duh -- addition is how you combine stuff. And yeah, you take the derivative to see how your position is changing. What else would you do?"

One can always dream. Happy math.

PS. Want more?

• I have another visual introduction to calculus in terms of shapes
• Learn to see integration as a better multiplication