Chain Rule & Implicit Differentiation: MATH 1211 Guide

Studocu is not sponsored or endorsed by any college or university Learning Guide Unit 4 -MATH 1211 Calculus (University of the People) Studocu is not sponsored or endorsed by any college or university Learning Guide Unit 4 -MATH 1211 Calculus (University of the People) Downloaded by Whitney turner (ct8682126@gmail.com) lOMoARcPSD|24613649

MATH 1211-01 Calculus - AY2023-T3  Dashboard  My courses  MATH 1211-01 - AY2023-T3  16 February - 22 February  Learning Guide Unit 4 Learning Guide Unit 4 Learning Guide Unit 4 Overview Unit 4: The Chain Rule and Implicit Differentiation Topics:  Chain Rule  Chain Rule with other derivative rules  Chain Rule with Trigonometry, Exponential and Logarithm Function Downloaded by Whitney turner (ct8682126@gmail.com) lOMoARcPSD|24613649

 Implicit Differentiation to find  Implicit Differentiation and equation of a tangent line Learning Objectives: By the end of this Unit, you will be able to:  Recognize the chain rule for a composition of three or more functions  Find the derivative of a complicated function by using implicit differentiation  Use implicit differentiation to determine the equation of a tangent line Tasks:  Peer assess Unit 3 Written Assignment  Read the Learning Guide and Reading Assignments  Participate in the Discussion Assignment (post, comment, and rate in the Discussion Forum)  Complete and submit the Written Assignment  Make entries to the Learning Journal  Take the Self-Quiz  Introduction Downloaded by Whitney turner (ct8682126@gmail.com) lOMoARcPSD|24613649

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  Image by ThisIsEngineering at pexels.com  We have seen the techniques for differentiating basic functions ( , , etc.) as well as sums, differences, products, quotients, and constant multiples of these functions. However, these techniques do not allow us to differentiate compositions of functions, such as, or Downloaded by Whitney turner (ct8682126@gmail.com) lOMoARcPSD|24613649

or . In this section, we study the rule for finding the derivative of the composition of two or more functions, called the chain rule. For example, if we take , then we split as , composition of two functions and , where and . Then we take the derivative of these two functions, f^' (x)=15 x^14 and u’(x)=2x. Finally, we finish by doing a composition of and and then multiply by .  So, we have, . This is chain rule.  For understanding the Implicit Differentiation, think about what we have done before. We have already studied how to find equations of tangent lines to functions and the rate of change of a function at a specific point. In all these cases we had the explicit equation for the function as and differentiated these functions explicitly. Suppose instead that we want to determine the equation of a tangent line to an arbitrary curve or the rate of change of an arbitrary curve at a point. In this section, we solve these problems by finding the derivatives of functions that define implicitly in terms of . For example, we might have a function like, , for Downloaded by Whitney turner (ct8682126@gmail.com) lOMoARcPSD|24613649

which we would need to find what that is the slope of the function at some point (say). Under this circumstance, we will use implicit differentiation. As you read through the material and solve the equations, important to memorize the process of the chain rule and implicit differentiation.  The information in this introduction was obtained from Openstax and the LibreText Project. LibreTexts content is licensed by CC BY-NC-SA 3.0 . Legal  Reading Assignment 1. Herman, E. & Strang, G. (2020). Calculus volume 1 . OpenStacks. Rice University.  Read Chapter 3, pages 290-299 and pages 312-319 o sections 3.6, and 3.8  Review the completed examples to understand the concepts and then complete some problems on your own. Review and practice will contribute to your success in this course.  You should attempt the problems in the book as indicated, and you are allowed to use the book and other resources to understand how to answer the problems. You are welcome to ask about these in the discussion forum. o 3.6 #’s 217, 223, 227, 234, 236, 241, 243, 256 o 3.8 #’s 301, 307, 311, 316, 323 Downloaded by Whitney turner (ct8682126@gmail.com) lOMoARcPSD|24613649

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2. The intuitive notion of the chain rule . (n.d.). GeoGebra Calculus Applets. http://webspace.ship.edu/msrenault/GeoGebraCalculus/derivative_intuitive_chain _rule.html  Play with the different functions here 3. Chain rule introduction [Video]. (n.d.). Khan Academy. https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-2-new/ab- 3-1a/v/chain-rule-introduction  Watch this brief introduction to the chain rule. 4. Implicit differentiation [Video]. (n.d.). Khan Academy. https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-2-new/ab- 3-2/v/implicit-differentiation-1  This video explains how leverage the chain rule to take the derivative implicitly. 5. Day, K., Sessions, D., Smith, L., & Wlazlo, A. (n.d.). Chain rule & implicit differentiation . Texas A&M University. https://www.math.tamu.edu/~shatalov/151H_Wlazlo%20Sessions %20Smith%20Day%20Math%20Project%20Implicit%20Chain.pdf  This slide show explains how the chain rule and implicit differentiation are used to differentiate complex equations. Discussion Assignment Downloaded by Whitney turner (ct8682126@gmail.com) lOMoARcPSD|24613649

In the discussion forum, you are expected to participate often and engage in deep levels of discourse. Please post your initial response by Sunday evening and continue to participate throughout the unit. You are required to post an initial response to the question/issue presented in the Forum and then respond to at least 3 of your classmates’ initial posts. You should also respond to anyone who has responded to you. Write an example of a function whose derivative can be found by using the following rules: a) Product rule and special function differentiation rules b) Power rule, quotient rule, and chain rule c) Chain rule twice d) Implicit differentiation and special function differentiation rule Your Discussion should be a minimum of 250 words in length and not more than 450 words. Please include a word count. Following the APA standard, use references and in-text citations for the textbook and any other sources. Written Assignment 1. Chains Inc. is in the business of making and selling chains. Let be the number of miles of chain produced after hours of production. Let be the profit as a function of Downloaded by Whitney turner (ct8682126@gmail.com) lOMoARcPSD|24613649

the number of miles of chain produced and let be the profit as a function of the number of hours of production. Suppose the company can produce 3 miles of chain per hour and suppose their profit on the chains is $4000 per mile of chain. Find and interpret (use complete sentences) each of the following (include units), , , and . How does relates to and ? 2. Use Desmos to graph the function and estimate the slope of the tangent line at (-1,1). Then find using implicit differentiation and plug in and . Compare and discuss the estimated slope with the slope you found analytically. 3. Let . Find in 3 different ways by following the instructions below in parts a, b and c: a) Develop the identity then take the derivative. Downloaded by Whitney turner (ct8682126@gmail.com) lOMoARcPSD|24613649

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b) View as and use the product rule to find . c) Apply the chain rule directly to the expression . d) Are your answers in parts a, b, c the same? Why or why not? 4. Find for the equation 5. Find the equation of the tangent line that passes through point (1,2) to the graph 6. Find for the equation 7. Find for the function 8. Find for the function 9. Find for the function Downloaded by Whitney turner (ct8682126@gmail.com) lOMoARcPSD|24613649

10. Find for the following functions: a) b) Learning Journal The Learning Journal is an evaluation tool designed to help you evaluate the importance of the material you learned but understand what you found relevant. In some cases, it should help you record problems you encountered but also how you resolved them. This does not mean you should resubmit what you already shared in the Discussions and assignments, but more relevant information. Learning Journals help you understand what information was presented and how you felt it interacted with your world. Downloaded by Whitney turner (ct8682126@gmail.com) lOMoARcPSD|24613649

Learning Journal prompt Consider one mathematical idea from the topics this week that you have found beautiful and explain why it is be must:  explain the idea in a way that could be understood by a classmate who has taken classes X and Y but has  address how this beauty is similar to or different from other kinds of beauty that human beings encounter. The Learning Journal entry should be a minimum of 400 words and not more than 750 words. Self-Quiz The Self-Quiz gives you an opportunity to self-assess your knowledge of what you have learned so far. The results of the Self-Quiz do not count towards your final grade, but the quiz is an important part of the University’s learning process and it is expected that you will take it to ensure understanding of the materials presented. Reviewing and analyzing your results will help you perform better on future Graded Quizzes and the Final Exam. Please access the Self-Quiz on the main course homepage; it will be listed inside the Unit . Downloaded by Whitney turner (ct8682126@gmail.com) lOMoARcPSD|24613649

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Checklist Peer assess Unit 3 Written Assignment Read the Learning Guide and Reading Assignments Participate in the Discussion Assignment (post, comment, and rate in the Discussion Forum) Complete and submit the Written Assignment Make entries to the Learning Journal Take the Self-Quiz Downloaded by Whitney turner (ct8682126@gmail.com) lOMoARcPSD|24613649

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