Product Rule and Quotient Rule
Product rule
Suppose that the derivatives and exist.
We calculate .
With the definition of derivative, we get
and letting will make this derivation a lot easier to follow.
Now we also have
which makes sense, given that is the difference of and .
Similarly, we let , and we get
Now we get
Subtracting on both sides gives
This can also be seen by calculating areas in the following picture
(although that doesn't work very well with negative numbers).
The total area is , and
subtracting the area of the rectangle results in the areas of other rectangles.

Now we get
If and exist, then
This is known as the product rule.
In general, and
are not the same.
For example,
but
In this situation, the product rule gives
which is the correct result.
Quotient rule
Suppose that and exists, and .
We calculate .
We have
By applying the product rule, we get
The power rule
gives ,
and by using that with the chain rule, we get
By bringing all this together, we get
If and exists, and , then
This is known as the quotient rule.
The quotient rule is messy and it doesn't simplify nicely.
Instead of using the quotient rule, you can always rewrite as
and then apply other derivative rules, because that's how the quotient rule is derived.
On the other hand, the quotient rule gets the job done in one step, although that step is ugly.
For example, you can use the quotient rule along with
power rule and derivative of sum to get
Without the quotient rule, the calculation looks like this
(see the derivation of the quotient rule for explanations):
Both of these results simplify to