MATH 234, Calculus for BusinessLecture 9, Textbook Sections 4.1, 4.2Derivatives of Sums, Differences, Products, QuotientsUniversity of Illinois, Urbana-ChampaignFebruary 20th, 2017Derivatives of Sums, Differences, Products, Quotients (University of Illinois, Urbana-Champaign)MATH 234, Calculus for Business February 20th, 2017 1 / 23AnnouncementsHW 4 is due Thursday, 2/23, 11:59pm; late HWs are not accepted.No quiz tomorrow, quiz next week.The scores and rubric for Exam 1 will be posted later tonight.You will receive your exam in section tomorrow.The curve isNEW = 76 +2434∗ (RAW − 66)If there are inconsistencies between the exam and the rubric, use the“Grade Explanation Form” to begin the “inconsistenciescheck”process. You have until next Wednesday to begin this process.During the “inconsistencies check” process, the entire exam will bere-examined, with focus given to the problems mentioned on the form.However, it is possible that the grade may go up or may go down.All “inconsistencies check” decisions are final.Derivatives of Sums, Differences, Products, Quotients (University of Illinois, Urbana-Champaign)MATH 234, Calculus for Business February 20th, 2017 2 / 23Review Definition of derivativeRemarkRecall from Lecture 7f0(a) = limh→0f (a + h) − f (a)hcan be understood in 3 ways:(1) the derivative of f (x) at x = a(2) the instantaneous rate of change of f (x) at x = a, or “the velocity”(3) the slope of the tangent line to the graph of f (x) at (a, f (a))RemarkRecall from Lecture 8, the derivative of f (x) isdfdx(x) =ddxf (x)= f0(x)| {z }different notations, same concept= limh→0f (x + h) − f (x)hDerivatives of Sums, Differences, Products, Quotients (University of Illinois, Urbana-Champaign)MATH 234, Calculus for Business February 20th, 2017 3 / 23Review ExampleConstant Rule: If f (x) = k, then f0(x) = 0Constant Multiple Rule: If f (x) = kg(x), then f0(x) = kg0(x)Sum Rule:ddxf (x) + g(x)=dfdx(x) +dgdx(x)Difference Rule:f (x) − g(x)0= f0(x) − g0(x)Power Rule: If f (x) = xn, then f0(x) = n ∗ xn−1for every nExampleFind the derivative of f (x) = 5x1/2+ 7x2/3+ 2x32Solutionf0(x) =5x1/2+ 7x2/3+ 2x320=5x1/20+7x2/30+2x320= 5x1/20+ 7x2/30+ 2x320= 512x12−1+ 723x23−1+ 232x32−1=52x−12+143x−13+ 64x31Derivatives of Sums, Differences, Products, Quotients (University of Illinois, Urbana-Champaign)MATH 234, Calculus for Business February 20th, 2017 4 / 23Section 4.2, Derivatives of Products and Quotients Initial questionFood For Thought:Recall that“the derivative of the sum is the sum of the derivatives”, and“the derivative of the difference is the difference of the derivatives”Is it true that“the derivative of the product is the product of the derivatives”, and“the derivative of the quotient is the quotient of the derivatives”?RemarkIf we replaced “derivative” with “limit”, then the statements are all true.Let us test the questions with some examples.Derivatives of Sums, Differences, Products, Quotients (University of Illinois, Urbana-Champaign)MATH 234, Calculus for Business February 20th, 2017 5 / 23Section 4.2, Derivatives of Products and Quotients Test the questions with examplesExampleLet f (x) = x5. Note that f (x) = x3∗ x2, a product of two functions.Is the derivative of the product is the product of the derivatives?SolutionBy the power rule, f0(x) = 5x4. This is the correct answer.If the derivative of the product is the product of the derivatives, thenf0(x) = (x3)0∗ (x2)0= (3x2) ∗ (2x) = 6x3which is wrong.The derivative of the product is NOT the product of the derivatives.Derivatives of Sums, Differences, Products, Quotients (University of Illinois, Urbana-Champaign)MATH 234, Calculus for Business February 20th, 2017 6 / 23Section 4.2, Derivatives of Products and Quotients Test the questions with examplesExampleLet f (x) = x5. Note that f (x) =x8x3, a quotient of two functions.Is the derivative of the quotient is the quotient of the derivatives?SolutionBy the power rule, f0(x) = 5x4. This is the correct answer.If the derivative of the quotient is the quotient of the derivatives, thenf0(x) =(x8)0(x3)0=8x73x2=83x5which is wrong.The derivative of the quotient is NOT the quotient of the derivatives.Derivatives of Sums, Differences, Products, Quotients (University of Illinois, Urbana-Champaign)MATH 234, Calculus for Business February 20th, 2017 7 / 23Section 4.2, Derivatives of Products and Quotients The Product RuleTheorem (The Product Rule)If f (x) = u(x) ∗ v(x), then f0(x) = u(x) ∗ v0(x) + u0(x) ∗ v(x).ExampleLet f (x) = x5. Note that f (x) = x3∗ x2, a product of two functions.Use the product rule to compute f0(x).SolutionBy the product rulef0(x) = x3∗ (x2)0+ (x3)0∗ x2= x3∗ (2x) + (3x2) ∗ x2= 2x4+ 3x4= 5x4This is consistent with the answer given by the power rule.Derivatives of Sums, Differences, Products, Quotients (University of Illinois, Urbana-Champaign)MATH 234, Calculus for Business February 20th, 2017 8 / 23Section 4.2, Derivatives of Products and Quotients The Product RuleTheorem (The Product Rule)If f (x) = u(x) ∗ v(x), then f0(x) = u(x) ∗ v0(x) + u0(x) ∗ v(x).ExampleLet f (x) = (2x + 1)(3x + 2). Compute f0(x) in two waysMultiply to get a quadratic function for f (x), and proceed as before.Use the product rule.Solution (First way: multiply and proceed as before)Multiplication gives f (x) = 6x2+ 7x + 2. Using sum rule, power rule, etcf0(x) = (6x2)0+ (7x)0+ (2)0= 6 ∗ (x2)0+ 7 ∗ (x)0+ 0= 6 ∗ (2x) + 7 ∗ 1 = 12x + 7Derivatives of Sums, Differences, Products, Quotients (University of Illinois, Urbana-Champaign)MATH 234, Calculus for Business February 20th, 2017 9 / 23Section 4.2, Derivatives of Products and Quotients The Product RuleTheorem (The Product Rule)If f (x) = u(x) ∗ v(x), then f0(x) = u(x) ∗ v0(x) + u0(x) ∗ v(x).ExampleLet f (x) = (2x + 1)(3x + 2). Compute f0(x) in two waysMultiply to get a quadratic function for f (x), and proceed as before.Use the product rule.Solution (Second way: use the product rule)f0(x) = (2x + 1) ∗ (3x + 2)0+ (2x + 1)0∗ (3x + 2)= (2x + 1) ∗ (3) + (2) ∗ (3x + 2)= (6x + 3) + (6x + 4) = 12x + 7Derivatives of Sums, Differences, Products, Quotients (University of Illinois, Urbana-Champaign)MATH 234, Calculus for Business February 20th, 2017 10 / 23Section 4.2, Derivatives of Products and Quotients The Product RuleTheorem (The Product Rule)If f (x) = u(x) ∗