Page 88 - Calculus Workbook For Dummies

72 Part III: Differentiation

Giving It Up for the Product

and Quotient Rules

Now that you’ve got the easy stuff down, I’m sure you’re dying to get some practice
with advanced differentiation rules. The product rule and the quotient rule give you the
derivatives for the product of two functions and the quotient of two functions, respec-
tively and obviously.
The product rule is a snap. The derivative of a product of two functions, (first)(second),
equals first l ^ second + ^h first second l .
^
h
^ h
h
The quotient rule is also a piece of cake. The derivative of a quotient of two functions,
h
^ h
^ firsth ^ first l ^ second - ^h first second l
h
, equals .
^ secondh ^ secondh 2
Here’s a good way to remember the quotient rule. Notice that the numerator of the
quotient rule looks exactly like the product rule, except that there’s a minus sign
instead of a plus sign. And note that when you read a product, you read from left to
right, and when you read a quotient, you read from top to bottom. So just remember
that the quotient rule, like the product rule, works in the natural order in which you
read, beginning with the derivative of the first thing you read.
For some mysterious reason, many textbooks give the quotient rule in a different form
that’s harder to remember. Learn it the way I’ve written it above, beginning with first l .
^
h
That’s the easiest way to remember it.

Q. d _ x 2 sinx = ? Q. d x 2 = ?
i
dx dx ^ sinxh
d x 2 sinx = _i x i sinx + _ x ^ i sinx l 2 l
2 l
2
A. dx _ ^ h h d x 2 _ x i ^ sinx - _ x ^ i sinx l
2
h
h
= 2 x sinx + x 2 cosx dx ^ sinxh = 2
A. ^ sinxh
2 x sinx - x 2 cosx
= 2
sin x
One more thing: I’ve purposely designed
this example to resemble the product rule
example, so you can see the similarity
between the quotient rule numerator and
the product rule.