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!
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