Lesson 9: The Product and Quotient Rule

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Section 3.2The Product and Quotient RulesMath 1aFebruary 22, 2008AnnouncementsProblem Sessions Sunday, Thursday, 7pm, SC 310Oﬃce hours Tuesday, Wednesday 2–4pm SC 323Midterm I Friday 2/29 in class (up to §3.2)OutlineThe Product RuleDerivation of the product ruleExamplesThe Quotient RuleDerivationExamplesMore on the Power RulePower Rule for nonnegative integers by inductionPower Rule for negative integersRecollection and extensionWe have shown that if u and v are functions, that(u + v ) = u + v(u − v ) = u − vWhat about uv ? Is it u v ?Try this with u = x and v = x 2. Then uv = x 3, so (uv ) = 3x 2.But u v = 1(2x ) = 2x .So we have to be more careful.Is the derivative of a product the product of thederivatives?NO!Then uv = x 3, so (uv ) = 3x 2.But u v = 1(2x ) = 2x .So we have to be more careful.Is the derivative of a product the product of thederivatives?NO!Try this with u = x and v = x 2.So we have to be more careful.Is the derivative of a product the product of thederivatives?NO!Try this with u = x and v = x 2. Then uv = x 3, so (uv ) = 3x 2.But u v = 1(2x ) = 2x .Is the derivative of a product the product of thederivatives?NO!Try this with u = x and v = x 2. Then uv = x 3, so (uv ) = 3x 2.But u v = 1(2x ) = 2x .So we have to be more careful.Work longer hours.Get a raise.Say you get a 25 cent raise inyour hourly wages and work 5hours more per week. Howmuch extra money do youmake?Mmm...burgersSay you work in a fast-food joint. You want to make more money.What are your choices?Get a raise.Say you get a 25 cent raise inyour hourly wages and work 5hours more per week. Howmuch extra money do youmake?Mmm...burgersSay you work in a fast-food joint. You want to make more money.What are your choices?Work longer hours.Say you get a 25 cent raise inyour hourly wages and work 5hours more per week. Howmuch extra money do youmake?Mmm...burgersSay you work in a fast-food joint. You want to make more money.What are your choices?Work longer hours.Get a raise.Document Outline
• Announcements
• The Product Rule
• Derivation of the product rule
• Examples
• The Quotient Rule
• Derivation
• Examples
• More on the Power Rule
• Power Rule for nonnegative integers by induction
• Power Rule for negative integers