



While a
few people do math for it's own sake (yikes!),



most
people just use math to help describe things that happen in real
life. 


Lots of
things can be described with expressions like 3X ^{2} + 2X
5, 


and we've
already talked about doing derivatives and integrals of stuff like
that. 


Sometimes
though the expression that fits what's going on 


looks
more like 2 ^{X}, where the variable is the exponent. 


In expressions
like this the variable is usually time, 


and
something is getting bigger or smaller as time passes. 





The big
deal about this variable as exponent form is 


that the amount of
change that happens depends
on how much of the stuff 


that's getting bigger or smaller is already
there. 





One good
example of this is a savings account. 


The
interest that's earned is the amount of change in the balance. 


The more
money in the account, the more interest that's added. 





Say you
find a savings account that pays 10% interest a year. 


This
being a great rate, you deposit $100. 


The first
year you get $10 in interest which you leave in the account. 


The
second year you get interest on the $110, so you get $11 interest. 


You keep
letting the interest stay in the account, 


and it keeps paying 10%
interest. 





In the
following five years the interest amounts are: 


12.10,
13.31, 14.64, 16.10, 17.72. 


Each year
the interest is 10% of the amount in the account. 





Bacteria
cultures in a serer pipe or bad restaurant grow the same way. 





Now when
math types write the equations to describe this stuff, 


they
could start with 2^{X} or 9^{X} or <any
number>^{X} that they wanted 


and
adjust the constants to make the graph work right for that
particular curve. 





Sooner or
later, we are
going to want to find a derivative or integral 


of whatever we
equation we came up with. 


So if 5 ^{X}
is easier to use than 8^{X} or anything else, we'd use it
right? 


Well math
types tried to figure out if there was some exponent
base, 


5, or 12,
or whatever that was easier to use than all the others. 


It was a
long shot, but what the hey, 


it was
ancient times before the internet and pizza delivery. 


What else
did they have to do? 





They
might have come up with a table of derivatives like this 


(no
you're not supposed to know how to figure these out yet) 





F(x) 
F'(x) 
1.50^{X} 
1.50^{X}
x
0.405465 
1.75^{X} 
1.75^{X}
x
0.559616 
2.00^{X} 
2.00^{X}
x
0.693147 
2.25^{X} 
2.25^{X}
x
0.810930 
2.50^{X} 
2.50^{X}
x
0.916291 
2.75^{X} 
2.75^{X}
x
1.011601 
3.00^{X} 
3.00^{X}
x
1.098612 
3.25^{X} 
3.25^{X}
x
1.178655 
3.50^{X} 
3.50^{X}
x
1.252763 
3.75^{X} 
3.75^{X}
x
1.321756 
4.00^{X} 
4.00^{X}
x
1.386294 






See it? 


It looks
like there might be a number between 2.5 and 2.75 (closer to 2.75) 


where the
derivative of (that number)^{X} = (that number)^{X}
times one! 


That
would mean that the derivative of (that number)^{X} would be
ITSELF! 


That
would mean that the integral of (that number)^{X} would be
ITSELF TOO! 


Hey, you
can't get any easier than that. 





So did
they ever find the magic number? 


Well, yes
and no ... 


They got
close. 


It turns
out to be an irrational number (like pi) that starts with 2.71828... 


Because
this number is SO important (and so long) we use a letter to stand
for it. 


The
letter is "e." 


So the
derivative of eX is eX, 


and the
integral of eX is eX 


When you
write these equations, 


don't
forget the little "dX" and "+ C" things that go
along with Calculus. 











Remember
that the "+ C" part is only needed when we do an
indefinite integral. 


If we
have actual numbers by our integral, we can lose the + C. 








Examples: 





The
general form of this is ... 








Or if you
really want to see something that looks nasty ... 








All that
last one means is that if we have some nasty thing for an exponent, 


the
derivative is the original function times the derivative of the
exponent. 





It's a
different looking kind of chain rule inside/outside thing. 





If that
explanation is too confusing, look again at the first two examples
above. 





All this
also means that if we are finding an integral 


of one of these
that has
a nasty exponent, 


we need
the derivative of the nasty exponent 


sitting next to the e^{(exponent)}
to stuff back in. 





Some
people are never satisfied, and want a proof. 


If you
don't want to see it, the page is over. 


I repeat,
this is a nasty proof. 


You
probably don't need it. 





I'd turn
back if I were you ... 





Still
here? 





OK, here
goes. 





First,
where does "e" come from? 





We were
talking about things that grow based on how big they are at the
moment, 


and we
said that this turned out to be an exponential term 


like 2^{X}
or in general a^{X}. 





In an
exponential GROWTH equation, the a would be the original amount, 


plus the
interest percent. 


In our
savings account example above, a would be 1.1. 


1 for
100% of the original amount, plus .1 for the 10% it grows each time
period. 


So it is
more understandable as what it is, you can rewrite a^{X} as: 











If
instead of calculating the interest once a year 


the bank calculates
it twice a year, 


the
interest percent would be half as much 


and the number of periods
would be doubled. 


If the
bank calculated interest once every three months, 


the
interest percent would be one fourth as much 


and the
number of periods would be four times as much. 


Interest
rate and time periods have an inverse relationship. 


If we
write this idea with one variable, call it "n", 


the
equation would look something like: 








Now in
real life, stuff like bacteria don't call "time out" every
now and then, 


take
roll, and then grow. 


They grow
pretty much continuously. 


That
means the time periods are extremely, extremely tiny. 


To try to
build this into an equation, we have to make the time period smaller 


and smaller
and
smaller until we have ... 





A
LIMIT PROBLEM!!! 


A limit Problem, why did it have to
be a limit problem. 


This one
is: 








Now if we
try to take the limit of the exponent first, we're in trouble! 


So we
need to get rid of the exponent. 


How? 


We
multiply it out! 


How? 


By using
a little trick from Algebra called the binomial theorem. 


It says
that if we have (a + b) ^{n}, which we really do, 


this
multiplies out to the form: 








As n goes
to infinity, the number of terms we get from this goes to infinity. 


Our only
hope, is that after the first few terms, 


the value
of each terms gets to small to matter. 





Well
guess what? 





That's
just what happens! 





Putting
in 1 and 1/n for a and b, we get: 








a bit of
combining gets us: 








canceling
n's 








now as n
approaches infinity, 


terms
with n in the denominator get closer and closer to zero. 


That
gives us: 





If you
really want to get technical ... 








Adding on
each additional 1/factorial term 


adds way less and less to the total
value. 


The total
of all the terms adds up to ... wait for it ... 2.71828... also
known as "e." 





Since the
number we get when we add the terms stays at about 2.71828... 


no matter
how many terms we add, 


the
series of terms is said to "converge" on that value. 


This
convergence is like an asymptote. 





So big
deal! e is the answer to some continuous growth function, 


but why
is it that e is it's own derivative???? 





OK. Here
goes. This is the big one ... 











Now,
going through the same drill we just did ... 








simplifying: 








canceling
n's ... 








take the
limit as n approaches infinity ... 








So we get
a string of terms for e^{X} ... 








Take the
derivative of this ... 








Drop the
leading zero from the derivative ... 








AND
WE GET THE ORIGINAL FUNCTION!!!!!!!!






copyright 2005 Bruce Kirkpatrick 
