by **E Mouse** » Sat Jun 10, 2006 2:15 am

The best explanation is probably the one that my Calc professors have been beating me over the head with for the past few years.

You know how you can graph a curved line on a 2-d peice of paper? Turn it into a proper x-and-y graph. Now find an equation that will always return the right y value for the graphed line if you give it the corresonding x value for that point (this isn't always easy, and only works if the line doesn't 'overlap' with two values for a given x). This is the function that defines that line.

Taking the derivative of a function gives you a seperate function in terms of x and y that can be used, at any point, to determine the rate of change of the original line/graph/function at that same point. Basically, it tells you if the line is going to the upper-right or lower-right at the point you solve it at, and how fast. (Derivatives are always defined going from left to right.) The value of the derivative at a point is called the slope of the original line (at that point). Slopes are only constant for straight lines.

Sometimes you can take the derivative of the derivative itself, and this second(-order) derivative can be used for less useful applications on the original equation, like concavity, but I doubt you care about that.

If Physics was more your strong point, here's an alternate description, and a useful application: the position of an object compared to time can be considered a basic graph. If you find the function for it and take the derivative, you get the function for its velocity. If you take the derivative of that, you get the acceleration formula.

So basically, the location of something at a point of time is the normal equation, before taking any derivatives. The derivative of that equation will give the speed of the object at any valid point. The second derivative of the position (derivative of velocity) gives how much the object is speeding up or slowing down.

Integrals are... basically derivatives in reverse, but more difficult to deal with. The integral of a function that only has points above the x-axis (i.e. positive y) will give the function to find the area under the curve. (If part of the original line is below the x-axis, this part will be considered 'negative area' and actually be SUBTRACTED from the 'area.' It's really weird, so they're kinda useless on their own.)

As for how to mathematically take derivatives and integrals, that's a lengthly thing to explain, since it's different depending on what functions you're playing with. Any triginometric functions screw things up (guess what the derivative of Secant is!), as do any non-basic functions multiplied/divided by one another. If you really want to know, take a night course in it or something. It's mildly interesting.

Of course, I probably screwed this up by being too detailed and taking too long to post, but oh well. <p>

<span style="font-size:xx-small;">"Their rhetoric... You didn't put communists in his bed did you!" came Amber's indignant reply.

"Why not? All I had to do was open a gate to his bed and stick up a sign saying 'Hot virgin willing to make the ultimate sacrifice in the name of international socialist fraternity.'"</span>

<span style="color:blue;font-size:xx-small;">Excaliburned:</span> <span style="font-size:xx-small;">Ah yes, I'm thinking of having the USS Bob be preserved outside the Arena as a monument of sorts</span></p>