Given a function , how would you differentiate the function?

Differentiating it with the definition would be quite a chore. Just try expanding ...

Well, is the product of two functions, and , right?

Now think about what the derivative represents. The derivative describes a rate of change. How does the product change?

If we increase by some small quantity , it will cause a change in as well as in . The changes in the composite functions in turn depend on and . So we'd expect the terms and to crop up somewhere.

Since we're dealing with a product, it'd also be logical if there was some multiplication going on here.

Thus, the rule for the derivative of a product is:

It appears all the time, so you should be able to give the right answer.

Since:

then we get: