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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.