Professor Elvis Zap

I'm a going to tell you all you need to know
To be the master rapper of the calculus show:
Now the first thing you do is to take the limit;
Yes that's right, that's how to begin it.
You don't need to use epsilon delta stuff,
Four simple rules are going to be enough:

The limit of a constant is still a constant
And the limit of a sum is the sum of the limits
The limit of a product is the product of the limits
And the limit of the quotient is the quotient of the limits
provided that the limit of the denominator is not zero.
This last condition is certainly necessary
for in this case you must be clever!

Now the thing upon which we will take the limit
is an object known as the Newton quotient
It's a simple object so will you repeat it:
Say ef of ex plus delta ex minus ef of ex over delta ex
Say ef of ex plus delta ex minus ef of ex over delta ex
Now this here limit is evavaluated as delta ex approaches zero
Given a function when you take this limit
The resulting quantity is called the derivative
But in general it is quite inefficient
To take the derivative by using the limit
So a few simple rules will help you in fact
Master the world of the calculus rap.

The power rule is very simple
using it will cure your pimples
See the derivative of ex to the en
is en times ex to the en minus one
en times ex to the en minus one
don't be frightened don't be scared
differentiate the function ex squared
Yes the derivative of ex to the two
is two times ex

And any old fat headed thick skulled rube
Can differentiate the function ex cubed
Yes the derivative of ex to the three
is three times ex to the two

The next of the rules that we will employ
Is the product rule so stop calling me "Boy!"
The derivative of how times doo
is do dee how plus how dee doo

Where the conotation of the notation dee
Is to differentiate, naturally.

The next of the rules is the golden rule
This here rule is called the chain rule
We apply it to the composite of functions
it is very much like peeling an onion
layer by layer oh yes it's a fun one:
Now suppose that a function ach of ex
can be written in the form ef of gee of ex
Then the derivative of ach of ex
is the derivative of ef at gee of ex
times the derivative of gee at ex
We often say dee why dee ex
is dee why dee you times dee you dee ex
where why is ef and you is gee
why is ef and you is gee
but it's alphabet soup if you want to ask me

Now the next of the rules ain't a rule for fools
this here rule is the quotient rule
Yes the derivative of hi over ho
is ho dee hi minus hi dee ho
over ho ho

And the next few functions will not make you go mental
these are the functions that are called transcendental
Say derivative of the sine of ex is
the cosine of ex.
And the derivative of the cosine of ex is
minus the sine of ex.
And the derivative of ee to the ex is
ee to the ex.
And the derivative of the natural log of ex is
one over ex.



Now put on your coat and put on your hat
Since you have completed the calculus rap
Yes put on your shoes and put on your pants
Because in the next song we do the calculus dance!