Chapter 6What's so special about Euler's number e?

"Who has not be amazed to learn that the function y=exy = e^x, like a phoenix rising again from its own ashes, is its own derivative?"

- Francois le Lionnais

I’ve introduce a few derivative formulas, but a really important one I left out was exponentials. So here I want to talk about the derivatives of exponential functions like 2x2^x, 7x7^x, and to show why exe^x is arguably the most important exponential.

Population Mass

First of all, to get an intuition, let’s focus on the exponential function 2x2^x.

And let’s think of that input as a time, tt, in days, and the output 2t2^t as a population size, perhaps of a particularly fertile band of π\pi creatures, which doubles every day.

Actually, instead of population size, which grows in discrete little jumps with each new baby π\pi creature, maybe let’s think of 2t2^t as the total mass of the population, which better reflects the continuity of this function 2t2^t.

For example, at time t=0t=0, the total mass 20=12^0=1, for the mass of one creature. At t=1t = 1 day, the population has grown to 21=22^1 = 2 creatures masses. And day t=2t = 2, it’s at 22=42^2 = 4, and in general it keeps doubling every day.

For the derivative, we want to find dMdt\frac{dM}{dt}, the rate at which this population mass is growing, thought of as a tiny change to mass divided by a tiny change in time. Let’s start by thinking of the rate of change over a full day, say between day 22 and day 33. In that case, it grows from 44 to 88, so that’s eight creatures masses added over the course of that day.

Notice, the rate of growth equals the population mass at the start of the day.

Between day 33 and day 44, the mass grows from 88 to 1616, so that’s a rate of 88 creature masses per day, which again, is equal to the population size at the start of the day. And in general, the rate of growth over a full day equals the population size at the start of that day.

Examples of the rate of change between days.

So it might be tempting to say that this means the derivative of 2t2^t equals itself; that the rate of change of 2t2^t at a given time tt equals the value of 2t2^t itself. On the day when 2t2^t is 88, the number of new creature-masses added per day is also 88; on the day when 2t2^t is 1616, the number of new creature-masses added per day is also 1616.

This is definitely in the right direction, but it’s not quite correct. What we’re doing here is making comparisons over the course of a full day, considering the difference between 2t+12^{t+1} and 2t2^t. But for the derivative, we need to ask about what happens for smaller and smaller changes. What’s the growth rate over the course of a tenth of a day... a hundredth of a day... one one billionth of a day, etc.

This is why I had us think of the function as representing population mass vs. time, since it makes sense to ask about a tiny change in mass over a tiny fraction of a day, but it doesn’t make as much sense to ask about the tiny change in the discrete population size per second.

We know that the derivative of the exponential function M(t)=2tM(t) = 2^t is close-to, but not quite equal-to itself, suggesting that the derivative of an exponential function is another exponential function. Looking at any point on the plot, will the actual derivative be smaller or larger than the output of the function?

More abstractly, for some tiny change in time dtdt, we want to understand the difference between 2t+dt2^{t+dt} and 2t2^t, all divided by dtdt. The change in the function per unit time, but now we’re looking very narrowly around a given point in time, rather than over the course of a full day.

Here’s the thing. I would love if there was some clear geometric picture that made what’s about to follow just pop out. Some diagram where you could point to one value and say "see that part, that is the derivative of 2t2^t". If you know of one, please, let me know. And while the goal here, as with the rest of the series, is to maintain a spirit of playful discovery, the type of play that follows will have more to do with finding numerical patterns, rather than visual ones.

Finding the Derivative

Start by taking a close look at the exponent term 2t+dt2^{t+dt} in the numerator. A core property of exponentials is that you can break this up as 2t2dt2^{t} \cdot 2^{dt}. That really is the most important property of exponents; if you add two values in the exponent, you can break up the output as a product of some kind. This is what lets you relate additive ideas, like tiny steps in time, to multiplicative ideas, like rates and ratios.

Applying this property gives us the expression below.

Notice how both expressions in the numerator contain the term 2t2^t, so we can factor out 2t2^t, leaving us with the expression below. And remember, the derivative of 2t2^t is whatever this whole expression on the right approaches as dtdt approaches 00.

At first glance this might look like an unimportant manipulation, but a tremendously important fact is that this term on the right in parentheses, where dtdt lives, is completely separate from the tt term itself; it doesn’t depend on the actual time where we started.

What happens if you plug in a very small value for dtdt into a calculator to approximate this term in parentheses?

Your answer:
Our answer:

Substutiting 0.0010.001 in for dtdt we get the value.

20.00110.001=0.693387462581\frac{2^{0.001}-1}{0.001} = \color{#4bb77e} 0.693 \color{grey} 387462581

Substutiting 0.00010.0001 in for dtdt we get the value.

20.000110.0001=0.693171203765\frac{2^{0.0001}-1}{0.0001} = \color{#4bb77e} 0.6931 \color{grey} 71203765

And, substutiting 0.000010.00001 in for dtdt we get the value.

20.0000110.00001=0.69314958282\frac{2^{0.00001}-1}{0.00001} = \color{#4bb77e} 0.69314 \color{grey} 958282

You’ll find that it for smaller and smaller choices of dtdt, this value approaches a specific number, around 0.693147...0.693147... So the derivative of 2t2^t is itself multiplied by this value.

Don’t worry if that number seems mysterious, the central point is that this is some kind of constant; unlike derivatives of other functions, all the stuff that depends on dtdt is separate from the value of tt itself.

So the derivative of 2t2^t is just itself, but multiplied by some constant. This should kind of make sense, because earlier it felt like the rate of growth for our population with size 2t2^t, at least when we were looking at changes over the course a full day. Evidently, the rate of change for this function over much smaller time scales is not quite equal to itself, but it is proportional to itself, with this very peculiar proportionality constant of 0.6931...0.6931...

Different Exponentials

There’s not much special to the number 22 here. If instead had we dealt with the function 3t3^t, the exponential property would also have led us to the conclusion that the derivative of 3t3^t is proportional to itself, but this time with a proportionality constant of about 1.09861.0986.

For any other base to your exponent you can have fun seeing what the various proportionality constants are, maybe trying to find a pattern.

Notice how the proportionality constant for the derivative changes from less than 11 for a base of 22 to greater than 11 for a base of 33.

For example, if you plug in 88 as the base of the exponential function, you can see the relevant proportionality constant is around 2.0792.079. And maybe, just maybe, you would notice that this number happens to be exactly 33 times the constant associated with the base of 22. So these numbers aren’t random, there is some pattern. But what is the pattern? What does 22 have to do with 0.69310.6931 and what does 88 have to do with 2.0792.079.

This figure illustrates different base 2 exponential functions.

A second question, which ultimately explains these mystery constants, is whether there is some base where that proportionality constant is 11; where the derivative of M(t)=atM(t) = a^t is not just proportional to itself, but equal to itself.

Euler's Number

As it turns out, there is a number. It’s the special constant ee, around 2.718282.71828, called Euler's number. In fact, it’s not just that ee happens to show up here, this is, in a sense, what defines the number ee . This special exponential function with Euler's Number as the base is called the exponential function.

All exponential functions are proportional to their own derivative, but the exponential function base ee alone is the special number so that the proportionality constant is 11, meaning ete^t actually equals its own derivative.

ddt(et)=et\frac{d}{dt} \left ( e^{t} \right ) = e^{t}

If you look at the graph of ete^t, it has the peculiar property that the slope of a tangent line to any point on the graph equals the height of that point above the horizontal axis.

Examples of the slope of the tangent line for the exponential function.

So how does the exponential function help us find the derivatives of other exponential functions? Well, maybe you noticed that different exponentials look like horizontally scaled versions of each other. This is true for all exponential functions, but best seen with exponential functions with related bases.

This means that you can re-write one exponential in terms of another's base. For example, if we have an exponential function of base 22 and want to re-write the function in terms of base 44, it can be written like this.

2x=412x2^{x} = 4^{\frac{1}{2} \cdot x}

One way to see how to convert between two bases is to zoom in on the graph between 00 and 11 to see how fast the first base grows to to the value of the second base. In this case, base 44 grows twice as fast as base 22 and reaches the output of 22 in half the time. So to convert base 44 to base 22 we can multiply the input tt of the base 44 function by the constant 12\frac{1}{2}, which is the same as scaling 4x4^x by a factor of 22 in the horizontal direction.

So we've found a function, the exponential function of base ee, with a really nice derivative property. Can we take any old exponential function and re-write it in terms of the exponential function? Or in other words, what constant do we multiply the input variable by to make the exponential function have the same output as another exponential function?

For example, let's try to re-write 2t2^t in terms of the exponential function.

ect=2te^{c \cdot t} = 2^{t}

As before, we can zoom in on a plot of the two functions, and compare their behavior. Specifically, how long does it take the exponential function to grow to 22?

Well, looking at the graph, it takes about t=0.693...t=0.693... units which is exactly equal to the same proportionality constant we found before! If we multiply the input variable tt in the exponential function by this constant, the exponential function has the same output as 2t2^t.

e(0.69314718056...)t=2te^{(0.69314718056...) \cdot t} = 2^{t}

This type of question we are asking leads us directly towards another function, the inverse of the exponential function, the natural logarithm function.

Natural logarithm

The existence of a function like this can answer the question of the mystery constants, and it’s because it gives a different way to think about functions that are proportional to their own derivative. There's nothing fancy here, this is simply the definition of the natural log, which asks the question "ee to the what equals 22".

e??=2e^{??} = 2

And indeed, go plug in the natural log of 22 to a calculator, and you’ll find that it’s 0.6931...0.6931..., the mystery constant we ran into earlier. And same goes for all the other bases, the mystery proportionality constant that pops up when taking derivatives and when re-writing exponential functions using ee is the natural log of the base; the answer the the question "ee to the what equals that base".

Importantly, the natural logarithm function gives us the missing tool we need to find the derivative of any exponential function. The key is to re-write the function and then use the chain rule. For example, what is the derivative of the function 3t3^t? Well, let's re-write this function in terms of the exponential function using the natural logarithm to calculate the horizontally-scaling proportionality constant.

3t=eln(3)t3^{t} = e^{\ln(3)t}

Then, we can calculate the derivative of eln(3)te^{\ln(3)t} using the chain rule by. First, take the derivative of the outermost function, which due to the special nature of the exponential funtion is itself. Then, second, multiply this by the derivative of the inner function ln(3)t\ln(3)^t, which is the constant ln(3)\ln(3).

ddt(eln(3)t)=eln(3)tln(3)\frac{d}{dt} \left( e^{\ln(3)t} \right) = e^{\ln(3)t} \cdot \ln(3)

This is the same derivative we found using algebra above, since ln(3)=1.09861228867...\ln(3) = 1.09861228867...

ddt(3t)=3tln(3)\frac{d}{dt} \left( 3^t \right) = 3^t \cdot \ln(3)

The same technique can be used to find the derivative of any exponential function.

d(at)dt=atln(a)\frac{d\left(a^{t}\right)}{d t}=a^{t} \ln (a)

In fact, throughout applications of calculus, you rarely see exponentials written as some base to a power tt. Instead you almost always write exponentials as ee raised to some constant multiplied by tt. It’s all equivalent; any function like 2t2^t or 3t3^t can be written as ecte^{c \cdot t}. The difference is that framing things in terms of the exponential function plays much more smoothly with the process of derivatives.

Why we care

I know this is all pretty symbol heavy, but the reason we care is that all sorts of natural phenomena involve a certain rate of change being proportional to the thing changing.

For example, the rate of growth of a population actually does tend to be proportional to the size of the population itself, assuming there isn’t some limited resource slowing that growth down. If you put a cup of hot water in a cool room, the rate at which the water cools is proportional to the difference in temperature between the room and the water. Or said differently, the rate at which that difference changes is proportional to itself. If you invest your money, the rate at which it grows is proportional to the amount of money there at any time.

In all these cases, where some variable’s rate of change is proportional to itself, the function describing that variable over time will be some exponential. And even though there’s lot’s of ways to write any exponential function, it’s very natural to choose to express these functions as ecte^{ct}, since that constant cc in the exponent carries a very natural meaning: It’s the same as the proportionality constant between the size of the changing variable and the rate of change.

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