The slope of a line is defined to be the rise divided by the run. For a
*curved* line, however, the slope has to be defined *at a
point*. Lines (curved or straight, but not infinitely steep) can always
be thought of as *functions* of a single variable. We call the
slope of a line evaluated at any given point its *derivative*, and
call the process of finding that slope *taking the derivative of the
function*.

Later we'll say a few words about multivariate (vector) differential calculus, but that is mostly beyond the scope of this course.

The definition of the derivative of a function is:

(86) |

This is the

First, note that:

(87) |

for any constant . The constant simply factors out of the definition above.

Second, differentiation is *linear*. That is:

(88) |

Third, suppose that
(the product of two functions). Then

(89) |

where we used the definition above twice and multiplied everything out. If we multiply this rule by we obtain the following rule for the differential of a product:

(90) |

This is a

We can easily and directly compute the derivative of a mononomial:

(91) |

or we can derive this result by noting that , the product rule above, and using induction. If one assumes , then

(92) |

and we're done.

Again it is beyond the scope of this short review to *completely*
rederive all of the results of a calculus class, but from what has been
presented already one can see how one can systematically proceed. We
conclude, therefore, with a simple table of useful derivatives and
results in summary (including those above):

(93) | |||

(94) | |||

(95) | |||

(96) | |||

(97) | |||

(98) | |||

(99) | |||

(100) | |||

(101) | |||

(102) | |||

(103) | |||

(104) | |||

(105) | |||

(106) |

There are a few more integration rules that can be useful in this course, but nearly all of them can be derived in place using these rules, especially the chain rule and product rule.