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