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Product and Quotient Rule
In the previous section we noted that we had to be careful when differentiating
products or quotients. It’s now time to look at products and quotients and see why.
First let’s take a look at why we have to be careful with products and quotients.
Suppose that we have the two functions
and
. Let’s start by computing the derivative of the
product of these two functions. This is easy enough to do directly.
Remember that on occasion we will drop the
(x)
part on the functions to simplify
notation somewhat. We’ve done that in the work above.
Now, let’s try the following.
So, we can very quickly see that.
In other words, the derivative of a product is not the product of the derivatives.
Using the same functions we can do the same thing for quotients.
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To differentiate products and quotients we have the
Product Rule
and the
Quotient
Rule
.
Product Rule
If the two functions
f(x)
and
g(x)
are differentiable (
i.e.
the derivative exist) then the product is
differentiable and,
The proof of the Product Rule is shown in the
Proof of Various Derivative
Formulas
section of the Extras chapter.
Quotient Rule
If the two functions
f(x)
and
g(x)
are differentiable (
i.e.
the derivative exist) then the quotient is
differentiable and,
Note that the numerator of the quotient rule is very similar to the product rule so be
careful to not mix the two up!
The proof of the Quotient Rule is shown in the
Proof of Various Derivative
Formulas
section of the Extras chapter.
Let’s do a couple of examples of the product rule.
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 Fall '08
 sc
 Quotient Rule

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