I've been wondering about something. As I understand it, the GPL license requires all derivatives (even if by inheritance) of software to be open source. Since it includes inheritance, library-type projects such as glibc (the GNU C library) and nusoap are often licensed under weaker licenses, such as the LGPL.

Anyway, so let's say somebody creates a PHP class, say, a captcha script. The author licenses it under the GPL, and releases it to the public. I come along, download this PHP captcha class, and use it in my website's registration area. Wouldn't I have therefore created a derivative work, and legally, wouldn't I have to release the source code (PHP) of my website?

it would only be that one captch script that would still be kept gpl, you are not obligated to give up source code of the rest of your site if you use the 'captcha script' using a gpl script does not mean that you need to give up all rights to your site, teh captch script, well, yes, but the rest of your site, no...its all yours

No, your site does not have to be released to the public. Only the code that you used in the captcha script. Upon request, you must give the licence and the link to the place you got it for it to be legal...but of course I don't forsee anyone ever emailing you asking for the licence of the captcha that you're using. Just make sure that you keep it on file.

I don't see this at all. If a user comes to your website and uses a captcha script, you wouldn't have to give them the source for it or your website.
I don't feel like reading the GPL again, so I'll go off of where I see a key difference. The GPL is dealing with people who are distributing software whereas a website is using GPL licensed software to those who use the site. Odds are it isn't technically correct by Stallman, but what do I care PHP is under an apache style license.

Since writing the original post, I've gone ahead and coded my own captcha script.

I am still curious about this whole issue though. Do you guys have any sources, or can you pinpoint a section of the GPL to support your statements? I've tried to look up stuff like this, but I still feel like it's open to interpretation.

This is what I found on http://www.gnu.org/licenses/gpl.html:

The GNU General Public License does not permit incorporating your program into proprietary programs. If your program is a subroutine library, you may consider it more useful to permit linking proprietary applications with the library. If this is what you want to do, use the GNU Lesser General Public License instead of this License.

I see two ways to interpret this snippet and the rest of the GPL for websites. For the first way, you can consider a custom script modified from a GPL script to only be in possession of one person, unless that person releases it, sells it, or distributes it in any manner. Even though thousands of users may visit it every day, the script is still only installed on a single owner's server. An analogy would be if I took the Linux kernel, made some fancy modifications, but did not distribute it in any manner. I can even let my friends use my computer with the modified Linux. Heck, I could probably charge people to use my computer or place billboards around the monitor. I am not obligated to release the source code to my modified Linux version because I did not release or distribute any other part of it.

The second way to interpret for websites it is to say that by using a custom script modified from a GPL script and placing it in a public website, the person has effectively distributed it (analogous to traditional program binaries), and therefore the modified source code is must therefore be released under the GPL. If the source code is not distributed, the website is essentially proprietary.

I see both interpretations as completely valid. Which one is correct? Or am I completely misunderstanding the whole thing and is there a third interpretation?

Well, I will try and add a thought. There is very little legal precedence or legislation in this arena, however, there are a few key distinctions.

1. Derivative Work - You are correct that the inclusion of a script in a website could be construed as a derivative work, or at a minimum the inclusion of a licensed and copyrighted material.

2. Distribution - the next question is whether a public website could be construed as distribution, and the answer is generally speaking no. The reason it is not considered distribution, is because it is not intended for the user to run or install the website, script, or program independent of your website, unless you had a "download source code" link or something similar for your site. Use is different from distribution.
And generally speaking, the short time a site is located in the volatile ram section of a computer is not considered distribution, although there is some indication in regards to emails and the like that it could be construed as storage.