Language name: Araneae
Derivative of: Nothing, There is literally no other language I can think of that this could be derived from. Maybe Sign Language, But even then, It has to be very different because Spiders don't have fingers.
Spoken by: Spiders! Specifically, Every species of giant spider in my world will "speak" some form of it.
As for language rules, There are obviously some strict ones here. Spiders don't have mouths that could regulate airflow to form spoken consonants and vowels (I am refusing to acknowledge TM spiders and MC spiders.) so that means we need to focus mostly on things such as click consonant combinations (A spider clicking its fangs together.) and a visual form of language using the spider's pedipalps (The things on the front of its cephalothorax that look like tiny legs.).
It will obviously be a very primitive language. That of an animal with the intelligence to know the value of communication.
Think of a Bee, Cat, Or Dog. They communicate in a simple way that gets the point across with as little effort as possible.
It'd also be nice for there to be a written form of some kind. So spiders could leave messages to other spiders. Ex. "Predator ahead." "Looking for a mate." Etc.
You know what, A great analogy for this language is Dark Souls, Where people communicate solely via gestures, And written messages on the ground.
This shouldn't be all that hard to create due to its simplicity, But it will be different to any other language you make on here.