Summary

Remember, these rules come right out of the limit definition of the derivative! It's essential to remember the limit definition.

For example, the product rule: if f(x)=u(x)v(x), and if u'(x) and v'(x) exist, then

How do we figure?

Now we do something really clever: we add the appropriate form of zero:

which, upon rearranging terms, gives

and distributing, produces

by the continuity of u (it has to be continuous to be differentiable!).

Example: #7, p. 234

The quotient rule is just a lot uglier to inspect using the definition of the derivative: the quotient rule states that, if f(x)=u(x)/v(x), and if u'(x) and v'(x) exist and if , then

``lo dee hi minus hi dee lo, over the denominator squared they go!''

This rule can be established via the product rule, in a clever way: #33, p. 234

Example: #20, p. 234

Example: #37, p. 235

Example: #42, p. 235