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What's so special about Euler's number e? | Chapter 5, Essence of calculus
3Blue1Brown
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May 12, 2026
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Transcript
0:14
I've introduced a few derivative formulas, but a
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really important one that I left out was exponentials.
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So here I want to talk about the derivatives of functions like 2 to the x, 7 to the x,
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and also to show why e to the x is arguably the most important of the exponentials.
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First of all, to get an intuition, let's just focus on the function 2 to the x.
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Let's think of that input as a time, t, maybe in days, and the output,
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2 to the t, as a population size, perhaps of a particularly
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fertile band of pie creatures which doubles every single day.
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And actually, instead of population size, which grows in discrete little jumps with each
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new baby pie creature, maybe let's think of 2 to the t as the total mass of the
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population.
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I think that better reflects the continuity of this function, don't you?
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So for example, at time t equals 0, the total mass is 2 to the 0 equals 1,
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for the mass of one creature.
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At t equals 1 day, the population has grown to 2 to the 1 equals 2 creature masses.
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At day t equals 2, it's t squared, or 4, and in general it just keeps doubling every day.
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For the derivative, we want dm dt, the rate at which this population mass is growing,
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thought of as a tiny change in the mass, divided by a tiny change in time.
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Let's start by thinking of the rate of change over a full day,
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say between day 3 and day 4.
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In this case, it grows from 8 to 16, so that's 8
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new creature masses added over the course of one day.
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And notice, that rate of growth equals the population size at the start of the day.
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Between day 4 and day 5, it grows from 16 to 32,
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so that's a rate of 16 new creature masses per day,
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which again equals the population size at the start of the day.
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And in general, this rate of growth over a full day
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equals the population size at the start of that day.
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So it might be tempting to say that this means the derivative
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of 2 to the t equals itself, that the rate of change of this
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function at a given time t is equal to the value of that function.
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And this is definitely in the right direction, but it's not quite correct.
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What we're doing here is making comparisons over a full day,
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considering the difference between 2 to the t plus 1 and 2 to the t.
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But for the derivative, we need to ask what happens for smaller and smaller changes.
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What's the growth over the course of a tenth of a day,
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a hundredth of a day, one one billionth of a day?
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This is why I had us think of the function as representing population mass,
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since it makes sense to ask about a tiny change in mass over a tiny fraction of a day,
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but it doesn't make as much sense to ask about the tiny change in a discrete population
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size per second.
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More abstractly, for a tiny change in time, dt,
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we want to understand the difference between 2 to the t plus dt and 2 to the t,
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all divided by dt.
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The change in the function per unit time, but now we're looking very narrowly
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around a given point in time, rather than over the course of a full day.
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And here's the thing, I would love if there was some very clear geometric picture
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that made everything that's about to follow just pop out,
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some diagram where you could point to one value and say, see, that part,
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that is the derivative of 2 to the t.
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And if you know of one, please let me know.
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And while the goal here, as with the rest of the series,
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is to maintain a playful spirit of discovery, the type of play that
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follows will have more to do with finding numerical patterns rather than visual ones.
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So start by just taking a very close look at this term, 2 to the t plus dt.
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A core property of exponentials is that you can
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break this up as 2 to the t times 2 to the dt.
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That really is the most important property of exponents.
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If you add two values in that exponent, you can
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break up the output as a product of some kind.
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This is what lets you relate additive ideas, things like tiny steps in time,
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to multiplicative ideas, things like rates and ratios.
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I mean, just look at what happens here.
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After that move, we can factor out the term 2 to the t,
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which is now just multiplied by 2 to the dt minus 1, all divided by dt.
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And remember, the derivative of 2 to the t is whatever
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this whole expression approaches as dt approaches 0.
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And at first glance, that might seem like an unimportant manipulation.
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But a tremendously important fact is that this term on the right,
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where all of the dt stuff lives, is completely separate from the t term itself.
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It doesn't depend on the actual time where we started.
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You can go off to a calculator and plug in very small values for dt here,
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for example, maybe typing in 2 to the 0.001 minus 1 divided by 0.001.
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What you'll find is that for smaller and smaller choices of dt,
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this value approaches a very specific number, around 0.6931.
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Don't worry if that number seems mysterious, the
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central point is that this is some kind of constant.
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Unlike derivatives of other functions, all of the stuff
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that depends on dt is separate from the value of t itself.
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So the derivative of 2 to the t is just itself, but multiplied by some constant.
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And that should make sense, because earlier it felt like the derivative for 2 to the t
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should be itself, at least when we were looking at changes over the course of a full day.
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And evidently, the rate of change for this function over much smaller
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timescales is not quite equal to itself, but it's proportional to itself,
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with this very peculiar proportionality constant of 0.6931.
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And there's not too much special about the number 2 here.
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If instead we had dealt with the function 3 to the t,
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the exponential property would also have led us to the conclusion that the derivative
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of 3 to the t is proportional to itself, but this time it would have had a
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proportionality constant of 1.0986.
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And for other bases to your exponent, you can have fun trying to see what the
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various proportionality constants are, maybe seeing if you can find a pattern in them.
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For example, if you plug in 8 to the power of a very tiny number,
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minus 1, and divide by that same tiny number, you'd find that
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the relevant proportionality constant is around 2.079.
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And maybe, just maybe, you would notice that this number happens
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to be exactly 3 times the constant associated with the base for 2.
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So these numbers certainly aren't random, there is some kind of pattern, but what is it?
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What does 2 have to do with the number 0.6931,
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and what does 8 have to do with the number 2.079?
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Well, a second question that is ultimately going to explain these mystery constants is
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whether there's some base where that proportionality constant is 1,
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where the derivative of a to the power t is not just proportional to itself,
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but actually equal to itself.
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And there is!
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It's the special constant e around 2.71828.
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In fact, it's not just that the number e happens to show up here,
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this is in a sense what defines the number e.
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If you ask why does e of all numbers have this property,
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it's a little like asking why does pi of all numbers happen
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to be the ratio of the circumference of a circle to its diameter.
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This is at its heart what defines this value.
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All exponential functions are proportional to their own derivative,
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but e alone is the special number so that proportionality constant is 1,
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meaning e to the t actually equals its own derivative.
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One way to think of that is that if you look at the graph of e to the t,
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it has the peculiar property that the slope of a tangent line to any point
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on this graph equals the height of that point above the horizontal axis.
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The existence of a function like this answers the question of the mystery constants,
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and it's because it gives a different way to think about functions
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that are proportional to their own derivative.
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The key is to use the chain rule.
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For example, what is the derivative of e to the 3t?
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Well, you take the derivative of the outermost function,
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which due to this special nature of e is just itself,
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and multiply by the derivative of that inner function 3t, which is the constant 3.
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Or rather than just applying a rule blindly, you could take this moment
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to practice the intuition for the chain rule that I talked through last video,
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thinking about how a slight nudge to t changes the value of 3t,
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and how that intermediate change nudges the final value of e to the 3t.
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Either way, the point is e to the power of some constant
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times t is equal to that same constant times itself.
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And from here, the question of those mystery constants
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really just comes down to a certain algebraic manipulation.
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The number 2 can also be written as e to the natural log of 2.
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There's nothing fancy here, this is just the definition of the natural log,
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it asks the question e to the what equals 2.
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So the function 2 to the t is the same as the function
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e to the power of the natural log of 2 times t.
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And from what we just saw, combining the fact that e to the t is its own
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derivative with the chain rule, the derivative of this function is proportional
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to itself, with a proportionality constant equal to the natural log of 2.
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And indeed, if you go plug in the natural log of 2 to a calculator,
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you'll find that it's 0.6931, the mystery constant we ran into earlier.
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And the same goes for all the other bases.
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The mystery proportionality constant that pops up when taking derivatives is just
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the natural log of the base. The answer to the question e to the what equals that base.
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In fact, throughout applications of calculus, you rarely
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see exponentials written as some base to a power t.
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Instead, you almost always write the exponential
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as e to the power of some constant times t.
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It's all equivalent, I mean any function like 2 to the t or
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3 to the t can also be written as e to some constant times t.
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At the risk of staying overfocused on the symbols here,
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I want to emphasize that there are many ways to write down any particular
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exponential function.
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And when you see something written as e to some constant times t,
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that's a choice we make to write it that way, and the number e is not fundamental to
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that function itself.
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What is special about writing exponentials in terms of e like this is
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that it gives that constant in the exponent a nice readable meaning.
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Here, let me show you what I mean.
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All sorts of natural phenomena involve some rate of
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change that's proportional to the thing that's changing.
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For example, the rate of growth of a population actually does
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tend to be proportional to the size of the population itself,
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assuming there isn't some limited resource slowing things down.
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And if you put a cup of hot water in a cool room,
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the rate at which the water cools is proportional to the difference in temperature
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between the room and the water, or said a little differently,
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the rate at which that difference changes is proportional to itself.
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If you invest your money, the rate at which it grows is
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proportional to the amount of money there at any time.
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In all of these cases, where some variable's rate of change is proportional to itself,
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the function describing that variable over time is going to
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look like some kind of exponential.
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And even though there are lots of ways to write any exponential function,
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it's very natural to choose to express these functions as e to the power
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of some constant times t, since that constant carries a very natural meaning.
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It's the same as the proportionality constant between
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the size of the changing variable and the rate of change.
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And as always, I want to thank those who have made this series possible.
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Thank you.
— end of transcript —
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