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### Course: AP®︎ Calculus BC (2017 edition) > Unit 4

Lesson 6: Derivatives of exponential functions- 𝑒 and compound interest
- 𝑒 as a limit
- Derivative of 𝑒ˣ
- Derivative of aˣ (for any positive base a)
- Derivatives of aˣ and logₐx
- Worked example: Derivative of 7^(x²-x) using the chain rule
- Differentiate exponential functions
- Differentiating exponential functions review
- Proof: The derivative of 𝑒ˣ is 𝑒ˣ
- Derivative of eᶜᵒˢˣ⋅cos(eˣ)

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# 𝑒 and compound interest

Sal introduces a very special number in the world of math (and beyond!), the constant 𝑒. Created by Sal Khan.

## Want to join the conversation?

- At9:20, shouldn't the yearly computation be 365.25 to account for leap years?(71 votes)
- Definitely it should not be! Even if it sounds clever from mathematical point of view, in real world it doesn't work this way. In fact, there are several methods how to measure a year in days. https://en.wikipedia.org/wiki/Day_count_convention#Actual_methods(6 votes)

- Why did Sal choose to use 100% interest? Couldn't this same principle be used/proven with different interest, for example 80% or 500%?(40 votes)
- I noticed the same thing, so I tried plugging in different numbers (x) for the numerator in the expression (1+x/n)^n, and found that this equals e^x as n->infinity. Kind of neat!(73 votes)

- How do you find n in the compound interest equation?(15 votes)
- Brilliant question! To find n, you need to use natural logarithm function.

Suppose you have a future value formula PV * (1+r)^n = FV where:

PV stands for present value;

FV stands for future value;

r stands for interest rate; and

n stands for a number of periods

So PV * (1+r)^n = FV can be rearranged to

(1+r)^n = FV/PV

Then we take natural logarithm ln

ln(1+r)n = ln(FV/PV)

Then we divide both sides by ln(1+r) and we get

n=(ln(FV/PV))/ln(1+r)

If you haven't learned about natural logarithms go to Logarithms playlist in the Algebra section.

If you are not very familiar with present value and future value formulas then the next playlist will cover Time Value of Money which is a very important concept.(30 votes)

- what's mean e?(15 votes)
`e`

stands for**Eulers's number**which was named after Swiss mathematician Leonhard Euler who found this irrational constant.

https://en.wikipedia.org/wiki/E_(mathematical_constant)(7 votes)

- Why do 12 months and 1 year have different percentages?(7 votes)
- That depends on interest calculation frequency.(12 votes)

- So if you borrow $1 at 100% annual interest compounded monthly for 2yrs. Would the answer be 1 ( 1 + 100%/24 ) ^ 24 or 1 ( 1 + 100%/12 ) ^ 24? I'm leaning towards the first but not sure.(14 votes)
- Yes, it's the first one.(0 votes)

- Reasons why not to go into debt…. To escape having to calculate all of this XD(8 votes)
- Compounded interest also applies to any account where you earn interests. In that case, you might want to do the calculations to learn how much you will earn.(5 votes)

- Why is
*e,*so special and magical as Sal describes it? Why don't we just say 2.7 rounded?(5 votes)- The same could be said about pi, which could just be called 3.14 rounded. It's just the way things are, I suppose.(10 votes)

- Why is e such a small number and how did they calculate e if it is infinite?(5 votes)
- e is indeed infinite; although we have calculated some of the digits does not mean that we calculated all infinite digits.

e is also a small number since if we keep putting on compound interest, your interest money will be more smaller every increment.

As there is an infinite amount of increments, the interest money will be increasingly minuscule and approach a certain sum of money: 2.71828... !(6 votes)

- Can someone simply this(2 votes)
- Sure, let's simplify the concepts of "e" and "compound interest."

1. ***e (Euler's Number)**:***:

"E" is a special mathematical constant known as Euler's number, denoted by the symbol "e." Its approximate value is approximately 2.71828. Euler's number is a fundamental constant that appears in various areas of mathematics, particularly in calculus, where it is the base for the natural logarithm.

The value of "e" is irrational, meaning it cannot be expressed as a simple fraction and its decimal representation goes on infinitely without repeating. It is an essential constant in various mathematical and scientific calculations and often arises in problems involving growth, decay, and rates of change.

One of the fundamental properties of "e" is the exponential function e^x, which has unique and useful mathematical properties, making it significant in many mathematical applications.

2. **Compound Interest

Compound interest is a method of calculating interest on a principal amount that includes not only the initial principal but also the interest that has accumulated over previous periods. In contrast, simple interest is calculated only on the initial principal amount.

The formula for calculating compound interest is:

A = P(1 + r/n)^(nt)

Where:

A = the future value of the investment/loan, including interest

P = the principal amount (initial investment/loan)

r = the annual interest rate (expressed as a decimal)

n = the number of times that interest is compounded per year

t = the number of years the money is invested or borrowed for

In the formula, "A" represents the final amount after "t" years with compound interest, which includes both the original principal and the accumulated interest. Compound interest is a powerful concept that allows investments to grow exponentially over time, as interest is continuously added to the principal, leading to increasing returns.

It's essential to be aware of compound interest when saving or investing, as it can significantly impact the growth of your money over long periods. On the other hand, when borrowing, compound interest can cause the total amount owed to increase more rapidly than simple interest would.

it dont get simpler then this(10 votes)

## Video transcript

Let's say that you are
desperate for a dollar. So you come to me the local loan shark, and you say hey I need
to borrow a dollar for a year. I tell you I'm
in a good mood, I willing to lend you that dollar
that you need for a year. I will lend it to
you for the low interest of 100% per year. 100% per year. How much would you have
to pay me in a year? You're going to have to pay the original principal what I lent
you plus 100% of that. Plus one other dollar.
Which is clearly going to be equal to $2. You
say oh gee, that's a lot to have to pay to pay back
twice what I borrowed. There's a possibility that
I might have the money in 6 months. What kind
of a deal could you get me for that Mr. Loanshark? I say oh gee, if your
willing to pay back in 6 months, then I'll
just charge you half the interest for half the time. You borrow one dollar,
so in 6 months, I will charge you 50% interest.
50% interest over 6 months. This, of course, was 1
year. How much would you have to pay? Well, you
would have to pay the original principal what you borrowed. The one dollar plus
50% of that one dollar. Plus 0.50, and that of
course is equal to 1.5. That is equal to $1.50. I'll
just write it like this. $1.50. Now you say well
gee that's I guess better. What happens if I don't
have the money then? If I still actually need a year. We actually have a system for that. What I'll do is just
say that okay, you don't have the money for me
yet. I'll essentially ... we could think about it.
I will just lend that amount that you need for
you for another 6 months. We'll lend that out. We'll lend that out for
another 6 months at the same interest rate at 50%
for the next 6 months. Then you'll owe me the
principal a $1.50 plus 50% of the principal, plus 75
cents. Plus 75 cents, and that gets us to $2.25. That equals $2.25. Another way of thinking
about it is to go from $1 over the first
period, you just multiply that times 1.5. If your
going to grow something by 50%, you just multiply it times 1.5. If your going to grow it
by another 50%, you can multiply by 1.5 again. One way of thinking about
it that 50% interest is the same thing as multiplying by 1.5. Multiplying by 1.5. If you start with 1 and
multiply by 1.5 twice, this is going to be the same thing. $2.25 is going to be 1
multiplied by 1.5 twice. 1.5 multiplied twice is the
same thing as 1.5 squared. You can see the same
thing right over here. This is the same thing.
100% is the same thing as multiplying by 2. As we
be multiplying 1 plus 1. This is multiplying by
2, so you could do this right over here. You could
do this as 1 times 1. 1 times 1 to the first ... I'm sorry
1 times 2 to the first power , because your only doing it over
one period over that year. You say once again
where's that 2? Well, if someone is asking for
100%, that means over the period you're going have to pay twice. You're going have to pay
the principal plus 100%. You're going have to pay
twice what you originally borrowed. If someone is
charging you 50% over every period, you're
going have to pay whatever you borrowed. That's
kind of the one part plus 50% of it. So 1.5 times what
you borrowed. You multiply times 1.5 every time.
If you wanted to see how this actually related to
the interest, you could view this as ... this
right over here is equal to 1 times, the interest
part is 1 plus 100% divided by 1 period to the first power. I know this seems like
a crazy way of rewriting what we just wrote over
here. Writing 1 plus 1, but you'll see that we
can keep writing this as we compound over different periods. This one right over here, we can rewrite. We can write as 1 times 1 plus 100%. Here we took our 100% for the year, and we divided into 2 periods.
Two 6 month periods. Each of them at 50%. 1
plus 100% over 2 is the same thing as 1.5, and
we compounded it over 2 periods. Let me do that
2 periods into a different color. The periods, let
me do in this orange color right over here.
You might start to see a pattern forming. Let's
say, well gee, I might have the money back in
... and you don't really like this. This is
$2.25. That was more than the original $2, so you
say, well what if we do this over every 12
months. I say, "Sure. We got a program for that." After every 12 months
... or after every month I should say, I'm just
going to charge you 100% divided by 12 interest.
This is equal to 8 1/3%. Having to pay back the
principal plus 8 1/3%, that's the same thing as
multiplying times 1.083 repeating. After 1 month you would
have to pay 1.083 repeating. After 2 months ... and
this isn't the scale that actually looks
more than 2 months, but it's not completely at
scale. After 2 months your going have to multiply by this again. Times 1.083 repeating,
and so that would get you 1.083 repeating squared.
If you went all the way down 12 months ... let
me get myself some space here. If you went all the way
down 12 months ... let me just. I should way from the
beginning 12 months, so another 10 months.
What's the total interest you would have to pay
over a year if you weren't able to keep coming up with the money? If you had to keep re-borrowing it. I kept compounding that interest. Well, you're going have
to pay 1.083 to the ... this is for 1 month. You
could view this as to the first power. This is
for 2 months, so you're going have to pay this to the 12th power. We have compounded over
12 periods, 8 1/3% over 12 periods. If you wanted
to write it in this form right over here,
this would be the same thing as the original principal. Our original principal
times 1 plus 100% divided by 12. Now we've divided
our 100% into 12 periods, and we're going to compound that 12 times. We're going to take
that to the 12th power. What is this going to
equal to? This buisness over here. We can get a
calculator out for that. I'll get my TI-85 out.
What is this going to be equal to? We could do it a couple of ways. This is 1.083 repeating.
Let's get our calculator out. We could do it a couple of
ways. Let me write it this way. Your going to get the same value.
I don't have to rewrite this one. I just did that there to
kind of hopefully you'd see the kind of structure
in this expression. 1 plus ... 100% is the same thing as 1. 1 divided by 12 to the 12th
power. 2.613, I'll just round. So approximately 2.613.
You say well this is an interesting game you all most forgot about your financial troubles,
and you're just intrigued by what happens if we keep going this. Here we compounded just
... we have 100% over here. Here we do 50%
every 6 months. Here we do a 12th of 100%, 8 1/3% every 12 months until we get to this number. What happens if we did every day? Every day. If I borrowed a one
dollar, and I'd say well gee I'm just going to ... each
day I'm going to charge you charge you one three hundred
sixty-fifth of a 100%. So, 100% divided 365,
and I'm going to compound that 365 times. You're
curious mathematically. You say well, what do we get then? What do we get after a year? You have your original
principal. Let me scroll over a little bit more to the
right, so we have more space. You're going to have
your original principal times 1 plus 100% divided
by not 12. Now we've divided the 100% into 365 periods. 365 periods. We're going to compound it. Every time we have to
multiply by 1 plus 100% over 365 everyday that
the loan is not paid. 365th power. You say oh gee taking
somebody to 365th power that's going to give me some huge number. Then you say well maybe
not so bad, because 100% divided by 365 is
going to be a small number. This thing is going to
be reasonable close to 1. Obviously, we can raise
1 to whatever power we want, and we don't get anything crazy. Let's see where this one goes. Let's see where this one goes. This is the same thing
as 1 plus. 100% is the same thing as 1 divided
by 365 to the 365th power. We get 2.71456. Let me put
it over here. Then we get ... This is approximately equal
to ... this approximate is a very precise
approximation, but 2.7 ... but my calculators
precision only goes so far. 2.7145675 and it keeps going on and on. This is really really
interesting. It looks as if we take larger
and larger numbers here, it just doesn't just
balloon into some crazy ginormous number. It seems
to be approaching some magical and mystical
number. It is, in fact, the case. That if you
would just take larger and larger ... if you were
to take your 100% and divide by larger and
larger numbers, but take it to that power, you're going to approach perhaps the most magical
and mystical number of all. The number E.
You can see it right over here in your calculator.
They have this E to the X. I can do that, so E to
the ... I'll raise it to the first power so you can look at the calculators internal representation of it. You see all ready raising
some ... doing 1 plus 1 over 365 to the 365th power, we got pretty ... we're starting
to get really really close to E. I encourage
you try this with larger and larger numbers, and your going to get closer and closer to this magical mystery. You almost wouldn't mind
paying the loan shark E dollars, because it's
such a beautiful number.