Product rule questions and answers pdf

Clues and fully worked model solutions/answers available free from mathsupgrade. PRODUCT RULE: To multiply when two bases are the same, write the base and ADD the exponents. All questions from past papers, specimen papers, sample papers are included. 6 Definition of the Definite Integral; 5. T O ZMDaLdpev owoirtrhv GIxnifdiZnliGttex pC]ahlhciuwlLu`sA. Show an example that proves your classmate wrong. Again, factored form will be nicer when we start setting these derivatives equal to zero and solving. Use the product rule and the basic rules for the exponential and logarithmic functions: (ex)0= ex and (lnx)0= 1=x. The simplified answer is given, but the exponent of one of . • Answer all questions. 3. four terms. 4x5. Nov 16, 2022 · Hint : Recall that with Chain Rule problems you need to identify the “inside” and “outside” functions and then apply the chain rule. The product rule Itis appropriatetouse thisrule when you want todifferentiate two functions which aremultiplied together. Use proper notation and simplify your final answers. At a school there are 88 girls and 70 boys. A restaurant has 5 starters and 16 main courses on its menu. For any real number a a and positive numbers m m and n n, the product rule for exponents is the following. 2/12/2015 10:24:48 AM. -. y + Δy = (u + Δu) (v + Δv) = uv + uΔv + vΔu + ΔuΔv. This unit illustrates this rule. Try them ON YOUR OWN first, then watch if you need help. n ] GATlblR ur[iugdhjtFsj drCeFsRedrDvxe`dN. This is often abbreviated to y′= uv′+u′v. Nov 21, 2019 · Click here for answers. d dx√625 − x2 = − x √625 − x2. You may also use any of these materials for practice. • Answer all questions and ensure that your answers to parts of questions are clearly labelled. Answers : Let y = (x 2 +7x+2) (x 3-log x) Use product rule and rewrite each expression as single exponent. Train 8th grade students to rewrite each exponential expression as a single exponent with this set of pdf worksheets. It turns out that Q is differentiable everywhere that g(x) ≠ 0. The log of a product is equal to the sum of the logs of its factors. How many different combinations are possible? 3. Don’t spend too long on one question. Information Section 3 The quotient rule The quotient rule is the last rule for di erentiation that will be discussed in these worksheets. Read each question carefully before you begin answering it. . View Jun 26, 2023 · Because quotients and products are closely linked, we can use the product rule to understand how to take the derivative of a quotient. 7. Multiply the number of items in each set. (b) Quotient Rule: 2 ' ' g f g fg g f dx d − = The numerator of the result resembles the product rule, but there is a minus instead of a EXPONENT RULES & PRACTICE. Matthew Daly. = (−3x3 − 4) ⋅ −csc 5x5cot Critical thinking question: 14) A classmate claims that (f ⋅ g)' = f ' ⋅ g' for any functions f and g. Again direct substitution gives 0 0 so use L’Hopital’s Rule a second time to give. These sample exam questions were originally included in the AP Calculus AB and AP Calculus BC Curriculum Framework, published in fall 2014. 1 Average Function Value; 6. Product Rule for Counting. The following is a list of worksheets and other materials related to Math 122B and 125 at the UA. Calculus Test Prep - 3. 6. For example, if you multiply a2 by p3, you cannot apply the product rule since the given bases are different. For instance, to find the derivative of f (x) = x² sin (x), you use the product rule, and to find the derivative of g (x) = sin (x²) you use the chain rule. Diagrams are NOT accurately drawn, unless otherwise indicated. (a) f(x) = (3x2 −5x)ex 100+ Chain rule problems for Bank exams - 1. The student will be given a two polynomials and be asked to find the derivative of those polynomials multiplied together by using the product rule. • The correct use of the product rule gives d dx x2 Differentiate. To find the derivative of any term, we do one the following rules: The Chain Rule. y = uv where u and v are differentiable functions of x. 2. 05t + 25, t ≥ 0. Product Rule : n√ ab = n√ a · n√ b : 3. Calculate … if order matters. 7 Computing Definite Integrals; 5. r Worksheet by Kuta Software LLC Oct 17, 2019 · Next: Product Rule for Counting Textbook Answers GCSE Revision Cards. S. There are a few rules that can be used when solving logarithmic equations. cos x 3. = 400 e–0. For example Rules of Radicals. Biden, Jr. You may nd it helpful to combine the chain rule with the basic rules of the exponential The product rule is the formal rule for differentiating problems where one function is multiplied by another. A special rule, the chain rule, exists for differentiating a function of another function. Then differentiate the. The chapter headings refer to Calculus, Sixth Edition by Hughes-Hallett et Step 1: We need to define the product rule. Find the value of t for which T = 300, giving your answer to 3 significant figures. V = ( p − qt )2 , t ≥ 0 , where p and q are positive constants, and t is the time in seconds, measured after a certain instant. 1. com Lecture 9: The product rule In this lecture, we look at the derivative of a product of func-tions. Your answer should contain only positive exponents. This is a set of chain rule, product rule and quotient rule differentiation questions for students to check their understanding (and/or recollection). After reading this text, and/or viewing Therefore, we can apply L’H^opital’s Rule. Download these Free Product Rules MCQ Quiz Pdf and prepare for your upcoming exams Like Banking, SSC, Railway, UPSC, State PSC. Warning • The product rule does NOT say that the derivative of a product is the product of the derivatives. Warm and bubbly qualified Physics and Maths teacher, with years of experience teaching in class and 1-to-1. 12 men working 8 hours per day complete a piece of work in 10 days. Practice Questions 1. It states that y = uv ) dy dx = u dv dx + v du dx: In other words you break up your function into two bits (your u and v) and Mar 19, 2024 · Chemistry MCQ Biology MCQ. Dec 29, 2020 · Recognize the pattern in our answer above: when applying the Product Rule to a product of three functions, there are three terms added together in the final derivative. (power, quotient, product, chain rule, exponential, logarithmic, trig and inverse trig) Refer to your packet sheets and book homework – finish these if you haven’t yet. This yields. • Fill in the boxes at the top of this page with your name. One of these rules is the logarithmic product rule, which can be used to separate complex logs into multiple terms. sin x. Since direct substitution gives 0 0 use L’Hopital’s Rule to give. Expressions like this take on the general form. June 30, 2021 . What will you learn from this book:- Prepare for the design interview — prepare for the design exercise and learn more about how tech companies hire product designers. Identify how many items to select from each set. 1) + x ( = 3 x. We will be able to compute so the derivative of Get help with your Product rule homework. ©m Z2B0\1e7o YKMuhtfaa USgoafBtjwVadrpel qLqLJCy. Ex: f = 4, g = 2x, − 2 x2 ≠ 0-2-Create your own worksheets like this one with Infinite Calculus. Ex: f = 2x, g = 4, 8 ≠ 0-2-Create your own worksheets like this one with Infinite Calculus. • Solution 1 . 1. In order to use the product rule for counting, we use the following steps: Observe and identify the number of sets to be selected. Name: Exam Style Questions. x 4 /: Simplify and explain. the result. Determine where A(t) = t2e5−t A ( t) = t 2 e 5 − t is increasing and decreasing. We desire a formula for Q ′ in terms of f, g, f ′, and g ′. The Product Rule for Counting Name: _____ Instructions • Use black ink or ball-point pen. You may select the number of problems, types of polynomials, and variable Jul 1, 2020 · The addition rule: \(P(\text{A OR B}) = P(\text{A}) + P(\text{B}) - P(\text{A AND B})\) Use the following information to answer the next ten exercises. y = uv = u(x) v(x). Remember the rule in the following way. In the 17th century Mathematicians were working on a range of problems and attempting to solve unanswered questions – as Mathematicians do! One notable problem was working with tangents to curves; the so-called Multiply If Eis an elementary matrix for a multiply rule with multiplier c6= 0 , then det(EA) = cdet(A). d. Compute the derivative of f(x) = x2√625 − x2. The product rule is if the two "parts" of the function are being multiplied together, and the chain rule is if they are being composed. 16. • Diagrams are NOT accurately drawn, unless otherwise indicated. The numerator of the result resembles the product rule, but there is a minus instead of a plus; the minus sign goes with the g′ g ′. Determine the value of p and the value of q . Again, try Chain Rule here, to see if you like that better! Note that factored form will be nicer when we get to chapter 4 (which is why I’m showing it here). Ensure you have: Pencil, pen, ruler, protractor, pair of compasses and eraser. • Answer the questions in the spaces provided Solutions to Exercises. For the example shown above, we have that . 2 Area Between Curves; 6. Guidance. where f and g are both differentiable functions. A little suffering is good for youand it helps you learn. The quotient rule is derived from the product rule and the chain rule; the derivation is given at the end of the worksheet for those that are interested. Differentiate . The rule of sum (Addition Principle) and the rule of product (Multiplication Principle) are stated as below. 15. The denominator is simply the square of the original denominator – no derivatives there. If n is a positive integer greater than 1 and both a and b are positive real numbers then, 1. Information The marks for each question are shown in brackets Product Rule Instructions • Use black ink or ball-point pen. King's College London - BSc Physics. 07 KB. Also, help them develop substantial skills in finding the value of the unknown exponent Chain Rule Squeeze Theorem Inverse Trig Implicit Differentiation Exponential and Logarithm Derivatives L’hopital You must know ALL the rules for finding derivatives. Hint. ©n W2H0S1_9u dKIu`tUan aS^oif_tRwCazrjei PL`L]Cm. Subtract the equation y = uv to get. Introduction. Product Rule Worksheets. However, this second expression is a limit of a continuous function, so we can just plug x Question 1(b): 720 Question 2: 24 Question 3: 420 Question 4(a): 10000 Question 4(b): 5040 Question 5: 40 Question 6: 699 Question 7: 5040 Question 8: 90 Question 9: 4000 Question 10: 4950 Question 11: 42 Apply Question 1: 13 Question 2: 8 Question 3: 1267 Question 4: 45 Question 5(a): 36 Question 5(b): 84 Question 5(c): 126 Author. The Ultimate Collection of ALL questions on PRODUCT RULE FOR COUNTING COMBINATIONS from EdExcel 1MA1 9-1 GCSE Mathematics - FREE. Solutions. log b (xy) = log b x + log b y. , 2) multiplied together. A curve, C, has equation y = tanx (a) Show that the turning points on C occur where tanx = —1 (b) Find an equation Of the tangent to C at the point where x = 0 A curve, C , has equation y = 3x2eX (a) Find , using the product rule for differentiation. n√ an = | a | if n is even. On June 30, 2021, President Joseph R. Jan 11, 2024 · Quotient Rule. p = 4, q = 1. Office of Air Quality Planning and Standards . = (−4x2 − 3) ⋅ −csc 3x4cot 3x4 ⋅ 12x3 + csc 3x4 ⋅ −8x dx = 4xcsc 3x4 ⋅ (12x4cot 3x4 + 9x2cot 3x4 − 2) dy. Solution 1: In this case we could proceed by multiplying out the product and then differentiating. Examiner. signed into law a joint resolution of Congress, adopted under the Congressional Review Act (“CRA”), disapproving the final rule and by chain rule dy dx = dy du du dx = 3eu (3x2 +1) = 3(3x2 +1)ex3+x: The Product Rule The product rule allows us to di˙erentiate a function when it is comprised of a product of two functions; i. docx, 20. f(x) =. As the ball cools, its temperature, T °C, minutes after it enters the liquid, is given by. So, applying L’H^opital’s Rule, we get lim x!0 sin(x) x L’H= lim x!0 cos(x) 1. Answer the questions in the spaces provided — there may be more space than you need. M [ vMda^dSek lwIi\tChx GIjnRfQionfiqtNeJ WCeaEl[cbuNlvufsI. Created Date: 10/8/2019 6:04:27 AM Product and quotient rules Introduction As their names suggest, the product rule and the quotient rule are used to differentiate products of functions and quotients of functions. Free trial available at KutaSoftware. Complete coverage of the topic. 5-a-day Workbooks The Product Rule for Counting Instructions Use black ink or ball-point pen. (axn) = naxn−1 dx. x 2 /: Simplify and explain. Since det(E) = 1 for a combination rule, det(E) = 1 for a swap rule and det(E) = cfor a multiply rule with multiplier c6= 0, it follows that for any elementary matrix Ethere is the determinant multiplication rule det(EA) = det(E)det(A): (d) Using your answer to part (b) , determine the nature of each of the stationary points. 2x. is a nice approach to simplification here. Inverse Property: n√ an = a if n is odd or. Qualified Teacher. Book Tutor. The rule follows from the limit definition of derivative and is given by. Before we get into differentiation questions and answers, some background (yes, you may skip past this intro!). • If pencil is used for diagrams/sketches/graphs it must be dark (HB or B). Solution. A heated metal ball is dropped into a liquid. Summary of the quotient rule. is defined as the derivative of f(x) multiplied by g(x). These Calculus Worksheets will produce problems that involve using product rule of differentiation. In the first term a = 4 and n = 2, in the second term a = 3 and n = 1 while the third term is a constant and has zero derivative. e. 1 x x lim − cos. • You must show all your working out. Answer: Staple products are those products that we always require at home or on the go. Whenever we do so, we will use a \L’H=" to denote that we have used the rule and \=" to denote our usual simpli cation. Definition If y = f (x) = u (x)·v (x) then y′= f′(x) = u (x)·v′(x)+u′(x)·v (x). Exercise 1(a) If y = 4x2 + 3x − 5, then to calculate its derivative with respect to x, we need the sum rule and also the rule that. We’ll solve this using three different approaches — but we encourage you to become comfortable with the third approach as quickly as possible, because that’s the one you’ll use to compute derivatives quickly as the course progresses. Now. The AP Calculus AB and AP Calculus BC Course and Exam Description, which is out now, includes that curriculum framework, along with a new, unique set of exam questions. Created Date. Your instructor might use some of these in class. Motivation Product rule Quotient rule Trigonometric functions Warning about the power rule Keep in mind: the power rule for d dx x n only holds when n is a number! Example: you might be tempted to say d • The product rule is used to di!erentiate a function that is the VCE Maths Methods - Chain, Product & Quotient Rules Di!erentiation questions (2) 7 5x. Questions and Answers . 8 Substitution Rule for Definite Integrals; 6. A much better method is to use the Product Rule. Applications of Integrals. 3 Substitution Rule for Indefinite Integrals; 5. ©A l2n0 41I4 5 tKbuet MaY KSuodfbtOw0a Orve U 0LhL RCc. Below is a walkthrough for the test prep questions. mc-TY-chain-2009-1. uk. 1) Answer key Product Rule Pre-algebra: E . Quotient Rule : Note that on occasion we can allow a or b to be negative and still have these properties work. q R TMHaUdMes gwwiotwh^ MIsnYfBirnvigtrei ACuaalHcuuJlcuLs\. aptitude. • the same answer 15. Environmental Protection Agency . It is an im-portant rule because it allows us to di erentiate many more functions. function. Differentiate these for fun, or practice, whichever you need. Answer. £75 / hour. Solve for the value of the missing exponent. For example, Rice, pulses, paper, pens, oil, soap, etc. It follows from the limit definition of derivative and is given by. The quotient rule helps to di erentiate functions like e2x x2, x2 PRODUCT RULE FOR COUNTING – PRACTICE QUESTIONS CALCULATOR ALLOWED 1. As long as -3 'B DC 3 —2/87 25 c 3 —3) 2 OC — 3 3 3 Pal N 7 OF . Do not use rules found in later sections. Math 122B - First Semester Calculus and 125 - Calculus I. Chain Rule Example #1. A Z iAhlWlX RrGiCgVhutzsk MrxeVsUe]rqvBeJdm. Jul 18, 2022 · Definition: The Product Rule for Exponents. It is a rule that states that the derivative of a quotient of two functions is equal to the function in the denominator g(x) multiplied by the derivative of the numerator f(x) subtracted from the numerator f(x) multiplied by the derivative of the denominator g(x), all divided by the Dec 21, 2020 · In this case it is probably simpler to multiply f(x) out first, then compute the derivative; here's an example for which we really need the product rule. 5x4. Quotient Rule: Find the derivative of y D : sin x sin x 4. com The quotient rule. The product rule is used when we want to differentiate the product of two functions. You must show all your working out. Δy = uΔv + vΔu + ΔuΔv. In the question, the first parentheses is f(x) and the second parentheses is g(x). Answers to Trig, Product and Quotient Rule Practice (ID: 1) dy. With this book you can practice this kind of mindset, prepare for job interview, learn how to interview other designers and find concepts for projects for your portfolio. e2x +ln(x2 +1). exercise-3. U. e O EAol7l6 jr 4i dgNh4t Js q FrXegs3exr Zvze id n. Nov 16, 2022 · 5. However this has a couple of drawbacks in that it would take quite a lot of work and also. Can't find the question you're looking for? Go ahead and submit it to our experts to be answered. File Size: 101 kb. 3 Volumes of Solids of Revolution / Method of Aug 10, 2022 · Introduction. Rule of Sum - Statement: If there are \ ( n\) choices for one action, and \ ( m\) choices for another action and the two actions cannot be done at the same time, then there are \ ( n+m\) ways to choose one of these actions In the same way as we have a rule for the product of functions, we also have a rule for the quotient of functions. ex lnx. MATH 171 - Derivative Worksheet. To complete the same work in 8 days, working 15 hours a day, the number of men required is. Let Q(x) be defined by Q(x) = f(x) / g(x), where f and g are both differentiable functions. The derivatives of functions such as y = f(x) = 2x sin(x) and y = f(x) = xex can be found using the product rule. Lecture 9: The product rule In this lecture, we look at the derivative of a product of func-tions. . Previous: Expanding Brackets using Pascal’s Triangle. In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature. Each terms contains only one derivative of one of the original functions, and each function's derivative shows up in only one term. am ⋅an = am+n a m ⋅ a n = a m + n. 39 KB. Q(x) = f(x) g(x), \eqquot1. KONICA MINOLTA bizhub PRESS 1052. Show Solution For this problem the outside function is (hopefully) clearly the exponent of 4 on the parenthesis while the inside function is the polynomial that is being raised to the power. V y uMRaad weh pw KiAth1 oI 4nlf 2i4n MiMtea 7Cna ElncXuWlnuqsZ. pdf, 309. Attempt every question. Product Rule: Find the derivative of y D . 5 Area Problem; 5. Simpson’s Rule Like any other approximation rule, Simpson’s works best when the interval is narrow and the function values over that interval have a similar shape to the approximation device (in this case, a quadratic curve). 11 years ago. So let. 04 KB. Here is a set of practice problems to accompany the Chain Rule section of the Derivatives chapter of the notes for Paul Dawkins Calculus I course at Lamar University. the chance of making a mistake is high. Critical thinking question: 13) Give a function that requires three applications of the chain rule to differentiate. QUOTIENT RULE: To divide when two bases are the same, write the base and SUBTRACT the exponents. the bases is missing. ©P H2A0h1O9` lKYuDtHaG `Sno[fXt[wlaPrleU DLCLACa. Example 3. File Type: pdf. Worksheets. The product rule is also called Leibniz rule named after Gottfried Leibniz, who found it in 1684. Question: 1. Find the temperature of the ball as it enters the liquid. Because quotients and products are closely linked, we can use the product rule to understand how to take the derivative of a quotient. Example 1: Compute for 2 4 ∙ 2 2. 2. Access the answers to hundreds of Product rule questions that are explained in a way that's easy for you to understand. When x changes by an increment Δx, these functions have corresponding changes Δy, Δu, and Δv. Here's a short version. You may use tracing paper if needed. In particular, let Q (x) be defined by. x 2 /. When t = 1 the volume of a soap bubble is 9 cm 3 and at that instant its volume is decreasing at the rate of 6 cm 3 per second. Download File. x 3 /. d dx(f g) = f′ ⋅ g − f ⋅g′ g2 d d x ( f g) = f ′ ⋅ g − f ⋅ g ′ g 2. File previews. ( x 4 + 20 x 3 + 100) is increasing and decreasing. • For example, what is wrong with the following derivation: d dx x2 = d dx x· d dx x= 1·1 = 1 ? • Answer: You are not using the product rule correctly. 2 The Product and Quotient Rules Example 1Find the derivative of the given function by using the appropriate rules of differentiation. Always start with the ``bottom'' function and end with the ``bottom'' function squared. dvi Created Date: 10/13/2009 2:54:38 PM Law of Exponents: Product Rule (a m *a n = a m+n) The product rule is: when you multiply two powers with the same base, add the exponents. 4. ZERO EXPONENT RULE: Any base (except 0) raised to the zero power is equal to one. This leaflet explains how. Thus, we can apply the product rule. Solution: We have expressions with the same bases (i. We have already computed. For example, the function is made up of two simpler functions of x. • Answer the questions in the spaces provided – there may be more space than you need. SEND. The quotient rule is a very useful formula for deriving quotients of functions. Remember that you cannot apply the product rule if the given bases are different. Age range: 16+. You’ll need the chain rule to evaluate the derivative of each term. 5 oryM — 20 C . com The product rule can extend to a product of several functions; the pattern continues – take the derivative of each factor in turn, multiplied by all the other factors left alone, and add them up. We will be able to compute so the derivative of The quotient rule is a formal rule for differentiating problems where one function is divided by another. −. Many answers: Ex. e) 3ex+c —20 Jun 30, 2021 · EPA’s 2020 Oil and Gas Policy Rule . This topic is included in Paper 1 for AS-level Edexcel Maths and Papers 1 & 2 for A-level Edexcel Maths. Solutions to Exercises. For the product rule you take the derivative of one function at a time while keeping the other constant and sum these together. Quotient Rule: Show that y D has a maximum (zero slope) at x D 0: x x sin x Feb 11, 2018 · Chain, Product, Quotient Rule Tough Recap! Subject: Mathematics. Aug 26, 2020 · pdf, 610. The headmaster is going to pick a boy and a girl at random. Home. Forty-eight percent of all Californians registered voters prefer life in prison without parole over the death penalty for a person convicted of first degree murder. (A) 4. Answer all questions. 2 x ) x ( h 9. The given answers are not simplified. Some function types, like exponentials, can cause problems because their shape over a broad . co. As for the product rule, we state the result. Each time, differentiate a different function in the product and add the two terms together. Step 2: We will first calculate and . Question 13: What is an Impulse Product? Answer: The product that a buyer feels compelled to buy after seeing it in a store is known as an Impulse product. The product rule says . Get Product Rules Multiple Choice Questions (MCQ Quiz) with answers and detailed solutions. 3. Note: Bases must be the same to use the product rule. How many different combinations of meals are possible? 2. 12. Resource type: Worksheet/Activity. The Leibniz rule gives us d/dx5 = 3x4 +2x4 = 5x4. If selecting two items from a set, Calculate n x (n - 1) …if order does not matter. + 2). 1) 2 m2 ⋅ 2m3 4m5 2) m4 ⋅ 2m−3 2m 3) 4r−3 ⋅ 2r2 8 r 4) 4n4 ⋅ 2n−3 8n 5) 2k4 ⋅ 4k 8k5 More Examples Function Derivative y = 5x7 dy dx = 57x7 1 = 35x6 y = 9 3 p x2 rst change to exponential notation y = 9x2=3 dy dx = 9 2 3 x2=3 1 = 6x 1=3 = 6 3 p x y = 3x6 4 + 2 3x 9 change to exponential notation We’ll illustrate in the problems below. 4 Lesson 19: Product Rules for Exponents > Part A: Product and Power Rules for Exponents > Explore Algebra 1 Student Worktext 19A EXPLORE Example 3 Find the missing number using the product of powers rule. Critical thinking question: 12) A classmate claims that (f g) ' = f ' g' for any functions f and g. Title: Microsoft Word - product-rule-easy Author: Nov 16, 2022 · Solution. = (−x2 + 2) ⋅ −csc2 4x5 ⋅ 20x4 + cot 4x5 ⋅ −2x dx = 2x(10x5 ⋅ csc2 4x5 − 20x3 ⋅ csc2 4x5 − cot 4x5) dy. 4 More Substitution Rule; 5. 3 Lets look at a few derivatives related to functions where we know the answer already but where we can check things using the product formula: • d dx (x3 ·x5) • d dx e3xe5x • d dx √ x/ √ x • d dx sin(x)cos(x) Before we look at the quotient rule which allows to differentiate f(x)/g The product rule is a formal rule for differentiating problems where one function is multiplied by another. In some cases it might be advantageous to simplify/rewrite first. Next: The Corbettmaths Practice Questions on Product Rule for Counting for Level 2 Further Maths. m x AANlSll IrDirgsh_tvs_ WrweFshefrdvfeJda. 1 cos x H lim − = lim x2 x→0 x→0. Other rules that can be useful are the quotient rule Correct answer: d dx[f(x)g(x)] = f′(x)g(x) +g′(x)f(x) Explanation: When two different functions ( f(x) and g(x)) are being multiplied together and we need to use what we call the product rule to find the derivative. Practice Questions. Many answers. Graduate. Note that the numerator of the quotient rule Title: mc-TY-quotient-2009-1. ah ya ud do ce mn jw vf kl ji