One important fact to keep in mind is that

The derivative of
`f`(`x`)
at `x`=`a` (or `f´`(`a`) )
is defined as

Also, if you have an implicitly
defined function between `x` and `y` like
`x`^{2} - 2 `x y` + `y`^{2}
= 1, then you can perform implicit differentation
(basically, just taking the derivative of everything with respect to
both `x` and `y` are tacking on `dx`s and
`dy`s to indicate which) to get 2`x dx` - 2 `x
dy` - 2 `y dx`
+ 2 `y dy` = 0. Then if you solve for `dy/dx`, this will
be equal (by the chain rule) to `y`´ and if you solve for
`dx/dy`, this will be equal to `x`´. Note
that in this case, either derivative will be in terms of both
`x` and `y`.

You may be wondering about the derivatives of your favorite trigonometric functions. Well,

- log(
`ab`) = log`a`+ log`b`

[This way logarithms turn multiplications into into additions was why log tables (and their analog cousins, slide rules) were used to do long multiplications before computers came along.] - log(
`a/b`) = log`a`- log`b` -
`e`^{(log x )}=`x`and log`e`^{x}= x - log
`x`log^{a}= a`x` -
`a`^{x}= e^{(x log a)} - (log(
`x`))´ = 1/`x` - log 1 = 0
Closely related to the natural logarithm is
the logarithm to the base
`b`, (log), which can be defined as log(_{b}x`x`)/log(`b`).Finally, derivatives can be used to help you graph functions. First, they give you the slope of the graph at a point, which is useful. Second, the points where the slope of the graph is horizontal (

`f`´(`x`) = 0) are particularly important, because these are the only points at which a relative minimum or maximum can occur (in a differentiable function). These points where`f`´(`x`)) = 0 are*called critical points*. To determine whether a critical point is a minimum or maximum, or more generally to determine the concavity of a function, second derivatives can be used;`f`´´(`x`) < 0 means a relative maximum/concave down,`f`´´(`x`) > 0 means a relative minimum /concave up. Finally, taking the limit as`x`goes to positive or negative infinity gives information about the function's asympotitic behavior. Towards that end, derivatives can help you out with some difficult limits: by L'Hôpital's rule, if lim`f`(`x`) and lim`g`(`x`) are both zero, then lim`f`(`x`)/lim`g`(`x`) = lim`f`´(`x`)/lim`g´`(`x`). The proof of L'Hôpital's rule relies on the Mean Value Theorem: that for any function`f`(`x`) differentiable between`a`and`b`, there is some point`c`between`a`and`b`such that the derivative of`f`at`c`is the same as the average slope between`a`and`b`:## Integrals

The integral of`f`(`x`) from`a`to`b`with respect to`x`is noted as

and gives the area under the graph of`f`and above the interval [`a`,`b`]. It can be defined formally as a Riemann sum: the limit of the areas of rectangular approximations to the area as the approximations get better and better.As stated before, integration and differentiation are inverse operations. To be precise, the fundamental theorem of calculus states that

More generally, using an application of the Chain Rule,

Knowing these facts, we now know a tremendous number of integrals: just flip the sides of any table of derivatives. Here are some further facts about integrals:

For`c`a constant,

The integral of a positive, continuous function from`a`to`b`with`b > a`is greater than zero. The integral from`a`to`b`of`f`(`x`)-`g`(`x`), with`f`(`x`) >`g`(`x`) in the interval [`a`,`b`] gives the area between`f`(`x`) and`g`(`x`).If those properties aren't enough to solve your integral, and if you can't find it in any table, then here are some further tricks of the trade:

- Substitution (the ``inverse'' of the Chain Rule):
- Integration by parts (the ``invserse'' of the Product Rule):
- Partial fractions: Every rational function with a denomminator
which can be broken up
into the sum of fractions of the form

where`A`,`B`,`C`,`D`and`E`are constants (of course not the same constants in the two forms) may be (more) easily integrated. - Numerical approximation. This may not give you give you an exact
answer, but approximating the area under
`f`(`x`) with rectangles, trapezoids, or even more complicated shapes can give you a value near the integral when no other method will work. For a brief explanation of the use of an application available to MIT students, see Definite Integrals on Maple, part of Using Maple for ESG Subjects, used as part of the MIT subject 18.01A at ESG.

gives the arclength of the graph of`f`(`x`) between`x`=`a`and`x`=`b`. The integral

gives the volume contained by revolving the graph of`f`(`x`) between`x`=`a`and`x`=`b`about the`x`-axis. The integral

gives the volume contained by revolving the graph of`f`(`x`) between`x`=`a`and`x`=`b`about the`y`-axis. Finally, the surface area of the surface formed by revolving the graph of`f`(`x`) between`x`=a and`x`=b about the`x`-axis can be found by the integral

Sometimes you will wish to take an integral over an unbounded interval such as from 1 to infinity, or to take an integral of a function that is undefined at some points (such as 1/`x`^{1/2}). These are improper integrals, and can be found by taking the limit of an integral over an interval that either grows towards infinity or towards the points where the function is undefined.Calculus Index Page | Back to the World Web Math Categories Page

watko@mit.edu Last Modifed January 10, 2000. - Substitution (the ``inverse'' of the Chain Rule):