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.
So, again we can see that,
To differentiate products and quotients we have the
Product Rule and the Quotient Rule.
Product Rule
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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 Product Rule is shown in the
Proof of Various Derivative Formulas section of the Extras chapter.
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