Section 3.4: Product and Quotient Rules
Note that the Power Rule from 3.3 claims to only work for positive
integers n. In 3.4 we learn that it works for any integer. In fact, it’s
actually true for any REAL number n.
Example: Let ( )
Find
( ).
We’ve seen that polynomials are easy to differentiate
because the derivative of a sum is the sum of the
derivatives, etc.
This doesn’t work so well for the product or quotient of
two functions.
For example, we can find the derivative of the following
functions:
But if we factor y, say, using
then …
Is it true that
=
?
But we can make this work for us. Since we know that
the coefficient of must be 24, what combinations do
we see in the coefficients below that would give us 24?
Similarly, since we know that the exponent of must be
23, what combinations do we see in the exponents below
that would give us 23?
So
:
Great. Why would we want to do a derivative that way
when it was easier to just use
?
Example: Let ( )
√
Find
( ).
We’ve seen this rule in this form:
If
, then
.
Officially:
Similarly, there must be a trick for finding the derivative
of a quotient, since:
But if we write y as
:
then we can see that
Try this with
.
Example: Find the equation of the line tangent to
( )
at x = 3.
Now that we’ve found
, we need to find
can also be written as
(
notation.)
)|
( ), which
. (Just another form of
Note: remember we saw that for ( )
( )=
Rule:
,
. We could verify this using the Quotient
Example: Find
.