The product rule is a very useful result from differential calculus:

A second rule that is somewhat harder to remember applies to quotients:

Strang has a beautiful proof of the quotient rule.

He starts by deriving a third one---the reciprocal rule. Start with this identity:

Use the product rule:

Now use the reciprocal rule and the product rule to derive the quotient rule:

Very pretty!

UPDATE: David Jerison uses the chain rule and the power rule to derive the reciprocal rule: