Product Rule:
Let y = f[g(x)], example, f = x^2, g = x+1 --> f[g(x)] = (x+1)^2
(Composition of functions)
dy/dx = f'[g(x)] * g'(x) where f' is the first derivative of
f.
Quotient Rule:
Let y = f(x)/g(x)
--> dy/dx = {[g(x) * f'(x)] - [f(x) * g'(x)]}/[g(x)]^2
First Principle:
Essentially, this is talking about the reasoning behind
differentiation. Basically, the idea comes from trying to plot a
straight line tangent to a point and taking its gradient as its
derivative. The idea of limits come into place into the derivation.
I now also cannot remember the definition of differentiation of
first principle off-hand. Need to go and check. 
Chain Rule:
Let y = f(x) * g(x)
--> dy/dx = f(x) * g'(x) + f'(x) * g(x)
Some guy called Lebiniz(sp?) showed this and this is also called
the Lebiniz's Rule.
The Product Rule, Quotient Rule and Chain Rule can all be derived
using First Principle. I don't think I want to go into that.
