How To Find Original Function From Derivative

How do y'all wish the derivative was explained to you lot? Here's my take.

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

$\displaystyle{ f'(x) =\lim_{dx\to 0} \frac{f(x+dx)-f(x)}{dx}}$

But what does information technology 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 hereafter prices, and purchase low / sell loftier to build your empire. You could fifty-fifty stop using the monkeys that pick random stocks for your portfolio.

Like this magic newspaper, the derivative is a crystal ball that explains exactly how a pattern will change. Knowing this, you can plot the past/present/hereafter, detect minimums/maximums, and therefore make amend decisions. That's pretty interesting, more than the typical "the derivative is the slope of a function" clarification.

Let'due south 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 gauge.

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

We all live in a shiny continuum

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

A line is made upwards of points? Sure.
And so there'due south an space number of points on a line? Yeah.
How do you cantankerous a room when in that location'southward an infinite number of points to visit? (Gee, thanks Zeno).

And even so, we move. My intuition is to fight infinity with infinity. Sure, there'south infinity points between 0 and 1. Simply I movement two infinities of points per second (somehow!) and I cross the gap in half a 2nd.

Altitude 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 tin compare rates ("How fast are you lot moving through this continuum?").

Information technology's strange, only you can see ten/5 every bit "I demand to travel 10 'infinities' in 5 segments of time. To do this, I travel 2 'infinities' for each unit of time".

Analogy: Run into division equally a rate of motion through a continuum of points

What's after goose egg?

Another encephalon-buster: What number comes subsequently zero? .01? .0001?

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

Even though we tin't calculate the number after nothing, it must exist there, right? Similar 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 large it is, but information technology's there!

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

Measurements depend on the instrument

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

Officer: Do you lot know how fast y'all were going?

Driver: I have no idea.

Officeholder: 95 miles per hour.

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

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

Non exactly. Imagine a video camera pointed at Clark Kent (Superman'southward modify-ego). The photographic camera records 24 pictures/sec (40ms per photograph) and Clark seems still. On a second-past-second basis, he'due south non moving, and his speed is 0mph.

Incorrect 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 merely he's really moving -- he goes too fast for our instruments!

Analogy: Similar a photographic camera watching Superman, the speed we measure depends on the musical instrument!

Running the Treadmill

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

Imagine a shirtless Santa on a treadmill (go on, I'll expect). We're going to measure out his heart charge per unit 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 exist his "under stress" heart rate, correct?

Nope. Encounter, 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 grade, if the electrodes weren't there, we wouldn't take a measurement.

What to do? Well, look at the organisation:

measurement = bodily amount + measurement effect

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

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

By the way, the "electrode effect" shows up everywhere. Research studies take the Hawthorne Event where people change their beliefs because they are beingness studied. Gee, it seems everyone nosotros scrutinize sticks to their diet!

Agreement the derivative

Armed with these insights, we can come across how the derivative models modify:

Derivative explanation plain english

Outset with some system to study, $f(x)$:

Change by the smallest amount possible ($dx$)
Get the before-and-later difference: $f(x + dx) - f(x)$
We don't know exactly how small $dx$ is, and we don't care: get the rate of motion through the continuum: $[f(x + dx) - f(x)] / dten$
This rate, even so small, has some fault (our cameras are besides slow!). Predict what happens if the measurement were perfect, if $dx$ wasn't in that location.

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

Limits: what happens when $dx$ shrinks to nothingness, across whatever 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 nosotros throw away $dx$ when it'southward not needed?".

My pet peeve: Limits are a mod formalism, they didn't exist in Newton's time. They assistance make $dx$ disappear "cleanly". But educational activity them before the derivative is like showing a steering wheel without a machine! It's a tool to aid the derivative work, not something to exist studied in a vacuum.

An Case: f(ten) = ten^2

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

$\begin{aligned} f'(x) &= \lim_{dx\to 0} \frac{f(x+dx)-f(x)}{dx} \\ &= \lim_{dx\to 0} \frac{(x+dx)^2-x^2}{dx} \\ &= \lim_{dx\to 0} \frac{x^2 + 2xdx + dx^2 - x^2}{dx} \\ &= \lim_{dx\to 0} 2x + dx \\ &= 2x \end{aligned}$

Notation the difference in the last two equations:

Ane has the error built in ($dx$)
The other has the "true" modify, where $dx = 0$ (we presume our measurements have no issue on the outcome)

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

i, four, ix, 16, 25, 36, 49, 64...

The absolute alter betwixt each upshot is:

1, three, 5, 7, 9, eleven, 13, fifteen...

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

Consider the jump from $x=2$ to $x=3$ ($3^ii - 2^two = v$). What is "five" made of?

Measured charge per unit = Actual Charge per unit + Mistake
$5 = 2x + dx$
$5 = two(2) + 1$

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

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

Simply in the existent world, measurements every 1.0 seconds is too tiresome. What if our $dx$ was 0.1? What speed would we measure out at $x=two$?

Well, nosotros examine the modify from $x=2$ to $10=2.1$:

$ii.i^two - 2^2 = 0.41$

Call up, 0.41 is what we inverse in an interval of 0.ane. Our speed-per-unit is 0.41 / .i = four.one. And once again we take:

Measured rate = Actual Rate + Error
$iv.1 = 2x + dx$

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

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

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

The derivative as "continuous partition"

I see the integral equally better multiplication, where y'all can use a changing quantity to some other.

The derivative is "ameliorate division", where y'all go the speed through the continuum at every instant. Something like 10/5 = ii says "y'all accept a abiding speed of ii through the continuum".

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

If you lot employ this changing speed to each instant (take the integral of the derivative), you recreate the original behavior, but like applying the daily stock market changes to recreate the full toll history. Merely this is a large topic for another solar day.

Gotcha: The Many meanings of "Derivative"

You'll see "derivative" in many contexts:

"The derivative of $x^two$ is $2x$" means "At every signal, 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 charge per unit of modify is 44." When $f(x) = x^ii$, 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 upwards. Sometimes people will say "derivative of $x$" and mean $dx$.

Gotcha: Our models may non be perfect

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

Some disobedient functions defy the prediction: there'south a departure betwixt 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 gauge, the derivative doesn't work on them because we can't actually predict their behavior.

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

Gotcha: Integration doesn't really exist

The relationship betwixt derivatives, integrals and anti-derivatives is nuanced (and I got it wrong originally). Hither'southward a metaphor. Beginning with a plate, your function to examine:

Differentiation is breaking the plate into shards. There is a specific process: accept a difference, find a charge per unit of modify, then assume $dx$ isn't there.
Integration is weighing the shards: your original function was "this" big. There's a process, cumulative addition, simply information technology 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.

At that place'south no algorithm to find the anti-derivative; we have to guess. We brand a lookup table with a bunch of known derivatives (original plate => pile of shards) and wait at our existing pile to run across if it's like. "Let'south find the integral of $10x$. Well, information technology looks like $2x$ is the derivative of $x^ii$. So... scribble scribble... 10x is the derivative of $5x^2$.".

Finding derivatives is mechanics; finding anti-derivatives is an art. Sometimes we become stuck: nosotros take the changes, apply them piece by piece, and mechanically reconstruct a pattern. It might not be the "real" original plate, but is practiced 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: information technology's the fundamental theorem of calculus! (It's similar saying "Aren't $a^2 + b^two$ and $c^ii$ the same? Yep, but this isn't obvious: it's the Pythagorean theorem!"). Thank you 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 nosotros can recite medieval German hymns). I demand the message behind the definitions.

My biggest aha! was realizing the transient role of $dx$: information technology makes a measurement, and is removed to make a perfect model. Limits/infinitesimals are a formalism, we tin'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 dissimilar sizes of infinity? (In some sense they're all "infinite", in other senses the range (0,1) is smaller than (0,ii))
What are the real rules about making $dx$ "go away"? (How do infinitesimals and limits really work?)
How practice nosotros 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.

How To Find Original Function From Derivative

We all live in a shiny continuum

What's after goose egg?

Measurements depend on the instrument

Running the Treadmill

Agreement the derivative

An Case: f(ten) = ten^2

The derivative as "continuous partition"

Gotcha: The Many meanings of "Derivative"

Gotcha: Our models may non be perfect

Gotcha: Integration doesn't really exist

Reading math

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