1.6 Differentiation Techniques Product and Quotient Rules
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Math 135 Business Calculus Spring 2009 Class Notes 1.6 Diff erent iatio n T echni ques: The Product and Quoti ent Rules In Section 1.5, we saw how to diff ere ntiate the sum of two functions. This sec tion sho ws how to diff erentiate the product or quotient of two functions. THE PRODUCT RULE While the derivative of the sum of two functions is the sum of the derivatives, the derivative of a product of two functions is not the product of the derivatives. The following is the rule for finding the derivative of a product. THEOREM 5 The Product Rule Let F (x) = f (x) · g(x). Then F 0 (x) = d dx £ f (x) · g(x) § = f 0 (x) · g(x) + f (x) · g 0 (x). EXAMPLE Find d dx £ (x 4 − 2x 3 − 7)(3x 2 − 5x) § . EXAMPLE Find d dx £ (x 3 − 3x 2 + x) 2 § . EXAMPLE The U.S. gross domestic product, in billions of dollars, can be approximated using the function P (t) = 567 + t(36t 0.6 − 104) where t is the number of years since 1960. a) Find the (instantaneous) rate of change P 0 (t). b) Find and interpret P 0 (45). 23
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Transcript of 1.6 Differentiation Techniques Product and Quotient Rules
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24 Chapter 1 Di ff erentiation
THE QUOTIENT RULE
Just as the derivative of a product of two functions is not the product of the derivatives, the derivativeof a quotient of two functions is not the quotient of the derivatives. The rule for diff erentiating quotientsis as follows.
THEOREM 6 The Quotient Rule
Let Q(x) =N (x)
D(x). Then
Q0(x) =d
dx
∑N (x)
D(x)
∏=
N 0(x) · D(x)−N (x) · D0(x)£D(x)
§2 .
EXAMPLE Diff erentiate f (x) =1 + x2
x3
a) By using the Quotient Rule.
b) By first rewriting the function as a sum of powers.
EXAMPLE Diff erentiate g(x) =
√x + 1
x2 + 2.
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