The Chain Rule is a specific case of the Product Rule. It looks at the situation, when you are taking the derivative of a function being multiplied by itself. In other words, when it is raised to a power other then 0. Lets see what happens when this occurs.
You should recognize this from a previous topic (Definition of the Derivative to be specific). Look at the derivative of x. As you can see it's 1. x is also the function being multiplied by itself (raised to a power). You should also know that by being able to complete this table you have unknowingly already used the Chain Rule. Don't believe me? Look at the definition of what the Chain Rule is. The Specific function being raised to a power in this case is x. The reason you can't see the Chain Rule taking place is because of the fact that the derivative of the function being raised to a power is 1. Repeat the same table, but raise x2 + 3 to those powers instead. Do you see the pattern yet? If you don't then repeat the table using a different function and repeat doing so until you DO see the pattern. The pattern you will eventually see can be represented by this:
Want more info? Check out these links:
Calculus SOS - the Chain Rule
Some practice problems
