# chain rule practice problems pdf

777.8 777.8 1000 1000 777.8 777.8 1000 777.8] >> Let’s solve some common problems step-by-step so you can learn to solve them routinely for yourself. ڹ�b� fx���f��6n�}��An�:p��q#����ΐ]?F�L�זM K�!�3���Yie�P����I�`ţJ��\V�5�%��)��u��g�E�*��X�lŦ��eL�����cq/��� �m���_�f����_Z���v� �a^�c*y�5m-�X�">�iY���L����#d85�_KH����5l��s����Xj�L?u�:b�0QM������+�Rx�&�B�ͥ�-��p^M�F���o1+Ay�S+���Ku��A���汦c�6/\Մz�o����0F��l�S�W�Q�#��h�#2�B'=�[�IH nCwl�`|�|� B�jX����Q��1����w�B��)���1g� ����&�2~+�@mE���� 7Q�QC4�\5۔�غ��2����e��I:�%������ŌJS �놉с�7*�^1װx�����M,�@�N��/0;�#���ԗ%վ6�"jI@$�9��� G�#���U��I;���4;(�eO���ƃqRhX�c��w)!a��T �C����[ZB��"�Y�g��-|�`/Η8���h��ѹ g������e'�e���$6�$�-��Τ�WuidH����ڰ,�\/�b�VF�Z�����V���,-���^�K8/gc$. 489.6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 611.8 816 >> /Widths[351.8 611.1 1000 611.1 1000 935.2 351.8 481.5 481.5 611.1 935.2 351.8 416.7 295.1 826.4 501.7 501.7 826.4 795.8 752.1 767.4 811.1 722.6 693.1 833.5 795.8 382.6 Online aptitude preparation material with practice question bank, examples, solutions and explanations. 611.1 798.5 656.8 526.5 771.4 527.8 718.7 594.9 844.5 544.5 677.8 762 689.7 1200.9 stream 589.1 483.8 427.7 555.4 505 556.5 425.2 527.8 579.5 613.4 636.6 272] Only in the next step do you multiply the outside derivative by the derivative of the inside stuff. Practice de-composing the following functions into two elementary functions f(x) ... chain rule, provided below for your convenience, ... As you do so, explain to yourself why the chain rule is the only approach that makes sense. 15 0 obj pdf doc ; Chain Rule - Practice using this rule. /Name/F1 /Widths[272 489.6 816 489.6 816 761.6 272 380.8 380.8 489.6 761.6 272 326.4 272 489.6 460.7 580.4 896 722.6 1020.4 843.3 806.2 673.6 835.7 800.2 646.2 618.6 718.8 618.8 /BaseFont/KCSLMJ+CMMI12 472.2 472.2 472.2 472.2 583.3 583.3 0 0 472.2 472.2 333.3 555.6 577.8 577.8 597.2 Hint : Recall that with Chain Rule problems you need to identify the “inside” and “outside” functions and then apply the chain rule. Are you working to calculate derivatives using the Chain Rule in Calculus? The power rule combined with the Chain Rule •This is a special case of the Chain Rule, where the outer function f is a power function. pdf doc ; Rules - Practice with tables and derivative rules in symbolic form. Read More. 761.6 489.6 516.9 734 743.9 700.5 813 724.8 633.9 772.4 811.3 431.9 541.2 833 666.2 Here are a set of practice problems for the Derivatives chapter of my Calculus I notes. /FontDescriptor 26 0 R If you notice any errors please let me know. ]l��G��Bj1UA0�}~u��Ơ"z��t���&�k�S1#�1MT4��b����LvBhiY�)-)��{�6�L�IUtYD�0:@3A~� ���l����$�W(Դ���h�mzX�ϊ�I���h�Oy. Use the chain rule to ﬁnd . ∂w. Chain Rule: Problems and Solutions. Solving Word Problems Involving Subtraction. Product & Quotient Rules - Practice using these rules. /FontDescriptor 20 0 R /BaseFont/XWRGUE+CMR12 If you're seeing this message, it means we're having trouble loading external resources on our website. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 663.6 885.4 826.4 736.8 Many answers: Ex y = (((2x + 1)5 + 2) 6 + 3) 7 dy dx = 7(((2x + 1)5 + 2) 6 + 3) 6 ⋅ 6((2x + 1)5 + 2) 5 ⋅ 5(2x + 1)4 ⋅ 2-2-Create your own worksheets like this one with Infinite Calculus. Covered for all Bank Exams, Competitive Exams, Interviews and Entrance tests. /BaseFont/LNKQLF+CMMI8 ©1995-2001 Lawrence S. Husch and University of … 299.2 489.6 489.6 489.6 489.6 489.6 734 435.2 489.6 707.2 761.6 489.6 883.8 992.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 761.6 489.6 In other words, when you do the derivative rule for the outermost function, don’t touch the inside stuff! 444.4 611.1 777.8 777.8 777.8 777.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Practice - Additional practice covering this section. Find the … 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. If you're seeing this message, it means we're having trouble loading external resources on our website. 1. 324.7 531.3 590.3 295.1 324.7 560.8 295.1 885.4 590.3 531.3 590.3 560.8 414.1 419.1 >> 761.6 679.6 652.8 734 707.2 761.6 707.2 761.6 0 0 707.2 571.2 544 544 816 816 272 /Subtype/Type1 /Widths[660.7 490.6 632.1 882.1 544.1 388.9 692.4 1062.5 1062.5 1062.5 1062.5 295.1 Chain rule Statement Examples Table of Contents JJ II J I Page2of8 Back Print Version Home Page 21.2.Examples 21.2.1 Example Find the derivative d dx (2x+ 5)3. Problem: Evaluate the following derivatives using the chain rule: Constructed with the help of Alexa Bosse. Find the … >> Call these functions f and g, respectively. Want to skip the Summary? Use the chain rule to ﬁnd . 27 0 obj /FirstChar 33 ( Recall that , which makes ``the square'' the outer layer, NOT ``the cosine function''. Then, y is a composite function of x; this function is denoted by f g. • In multivariable calculus, you will see bushier trees and more complicated forms of the Chain Rule where you add products of derivatives along paths, /BaseFont/KNAEYV+CMSY8 If you are viewing the pdf version of this document (as opposed to viewing it on the web) this document contains only the problems themselves and no solutions are included in this document. /Name/F4 Diﬀerentiation: Chain Rule The Chain Rule is used when we want to diﬀerentiate a function that may be regarded as a composition of one or more simpler functions. w��. 531.3 826.4 826.4 826.4 826.4 0 0 826.4 826.4 826.4 1062.5 531.3 531.3 826.4 826.4 Answer: We apply the chain rule… 24 0 obj 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 295.1 >> %PDF-1.2 This rule is obtained from the chain rule by choosing u = f(x) above. We assigned plenty of MML problems on this section because the computations aren’t much di↵erent than ones you are already very good at. endobj Read More. Chain Rule worksheet MATH 1500 Find the derivative of each of the following functions by using the chain rule. 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 295.1 295.1 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 777.8 777.8 0 0 << << 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 606.7 816 748.3 679.6 728.7 811.3 765.8 571.2 << PRACTICE PROBLEMS: 1. The chain rule is a rule for differentiating compositions of functions. In fact, this problem has three layers. >> /FirstChar 33 Practice Problems with Fractions. /FirstChar 33 /FontDescriptor 29 0 R /Subtype/Type1 /LastChar 196 /LastChar 196 Problems: Chain Rule Practice One application of the chain rule is to problems in which you are given a function of x and y with inputs in polar coordinates. 805.5 896.3 870.4 935.2 870.4 935.2 0 0 870.4 736.1 703.7 703.7 1055.5 1055.5 351.8 Video lectures to prepare quantitative aptitude for placement tests, competitive exams like MBA, Bank exams, RBI, IBPS, SSC, SBI, RRB, Railway, LIC, MAT. Chain Rule Practice Problems Calculus I, Math 111 Name: 1. Solving Word Problems Involving Subtraction. /Type/Font 767.4 767.4 826.4 826.4 649.3 849.5 694.7 562.6 821.7 560.8 758.3 631 904.2 585.5 826.4 295.1 531.3] 3 0 obj << /Name/F6 481.5 675.9 643.5 870.4 643.5 643.5 546.3 611.1 1222.2 611.1 611.1 611.1 0 0 0 0 Chain Rule: The General Power Rule The general power rule is a special case of the chain rule. (a) F(x) = 4 p 1 + 2x+ x3 (b) g(t) = 1 (t4 + 1)3 (c) y= cos(a3 + x3) where ais a constant. 351.8 611.1 611.1 611.1 611.1 611.1 611.1 611.1 611.1 611.1 611.1 611.1 351.8 351.8 << /FirstChar 33 %���� 1444.4 555.6 1000 1444.4 472.2 472.2 527.8 527.8 527.8 527.8 666.7 666.7 1000 1000 Chain Rule problems or examples with solutions. /FontDescriptor 11 0 R /LastChar 196 462.4 761.6 734 693.4 707.2 747.8 666.2 639 768.3 734 353.2 503 761.2 611.8 897.2 In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature. /Widths[609.7 458.2 577.1 808.9 505 354.2 641.4 979.2 979.2 979.2 979.2 272 272 489.6 413.2 590.3 560.8 767.4 560.8 560.8 472.2 531.3 1062.5 531.3 531.3 531.3 0 0 0 0 Review your understanding of the product, quotient, and chain rules with some challenge problems. 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. 351.8 935.2 578.7 578.7 935.2 896.3 850.9 870.4 915.7 818.5 786.1 941.7 896.3 442.6 /Type/Font 947.3 784.1 748.3 631.1 775.5 745.3 602.2 573.9 665 570.8 924.4 812.6 568.1 670.2 /FirstChar 33 The general power rule states that this derivative is n times the function raised to the (n-1)th power times the derivative of the function. 495.7 376.2 612.3 619.8 639.2 522.3 467 610.1 544.1 607.2 471.5 576.4 631.6 659.7 777.8 777.8 1000 500 500 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 13) Give a function that requires three applications of the chain rule to differentiate. This unit illustrates this rule. 324.7 531.3 531.3 531.3 531.3 531.3 795.8 472.2 531.3 767.4 826.4 531.3 958.7 1076.8 (easy) Find the equation of the tangent line of f(x) = 2x3=2 at x = 1. endobj /Widths[1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 1000 1000 1000 777.8 /Subtype/Type1 756.4 705.8 763.6 708.3 708.3 708.3 708.3 708.3 649.3 649.3 472.2 472.2 472.2 472.2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 753.7 1000 935.2 831.5 Click HERE to return to the list of problems. Here are some example problems about the product, fraction and chain rules for derivatives and implicit di er-entiation. /Length 2498 384.3 611.1 675.9 351.8 384.3 643.5 351.8 1000 675.9 611.1 675.9 643.5 481.5 488 Solution We begin by viewing (2x+5)3 as a composition of functions and identifying the outside function f and the inside function g. 708.3 708.3 826.4 826.4 472.2 472.2 472.2 649.3 826.4 826.4 826.4 826.4 0 0 0 0 0 A few are somewhat challenging. 652.8 598 0 0 757.6 622.8 552.8 507.9 433.7 395.4 427.7 483.1 456.3 346.1 563.7 571.2 272 272 489.6 544 435.2 544 435.2 299.2 489.6 544 272 299.2 516.8 272 816 544 489.6 384.3 611.1 611.1 611.1 611.1 611.1 896.3 546.3 611.1 870.4 935.2 611.1 1077.8 1207.4 By using the Chain Rule an then the Power Rule, we get = = nu;1 = n*g(x)+;1g’(x) 720.1 807.4 730.7 1264.5 869.1 841.6 743.3 867.7 906.9 643.4 586.3 662.8 656.2 1054.6 9 0 obj %PDF-1.4 531.3 531.3 413.2 413.2 295.1 531.3 531.3 649.3 531.3 295.1 885.4 795.8 885.4 443.6 << That material is here. (Section 3.6: Chain Rule) 3.6.2 We can think of y as a function of u, which, in turn, is a function of x. 666.7 666.7 666.7 666.7 611.1 611.1 444.4 444.4 444.4 444.4 500 500 388.9 388.9 277.8 761.6 272 489.6] 597.2 736.1 736.1 527.8 527.8 583.3 583.3 583.3 583.3 750 750 750 750 1044.4 1044.4 /LastChar 196 >> Calculus Exam - Chain Rule & Implicit Practice Exam Solutions For problems 1-5, find the derivative. << The Chain Rule mc-TY-chain-2009-1 A special rule, thechainrule, exists for diﬀerentiating a function of another function. pdf doc ; CHAPTER 3 - Rules For Differentiation. x��Z�r�F��+x�)۽��c6'��\bݢY�T�R�'���4g8ZR��5$��� !�����i�a�7����w�n�����o[%��ϻk�e7_�����?n�������h�� k~�z����ǸL �A�MB�r�� ��n�>J=ަw���t�������p6�7������o˻����}����n>������wZ�O\��!I�����OZ��j����fJ4-�&�F�m�����?��7oec��dF�ֵ(ʜ��*J��~tE�@D'��=��0 (e�z,� �m[)��]l�+0m��( A@�� /Subtype/Type1 A.P. /Name/F3 Practice problems for sections on September 27th and 29th. It states: if y = (f(x))n, then dy dx = nf0(x)(f(x))n−1 where f0(x) is the derivative of f(x) with respect to x. /Name/F8 x��ZKo�F��Wpou����\f��n�ٍsJr�e��-z�����S�&�&դ(�2H0��&[Ů������櫯�I�$Bj��>$���I���j���'?��Xg�f�F��=����~���Ū���+����o��N%�:�4�#J�d��nIf��Pv�k+��W�~���� c�!�BRK��%K! Chain Rule Worksheets with Answers admin October 1, 2019 Some of the Worksheets below are Chain Rule Worksheets with Answers, usage of the chain rule to obtain the derivatives of functions, several interesting chain rule exercises with step by step solutions and quizzes with answers, … /Name/F5 /LastChar 196 Problems: Chain Rule Practice One application of the chain rule is to problems in which you are given a function of x and y with inputs in polar coordinates. 380.8 380.8 380.8 979.2 979.2 410.9 514 416.3 421.4 508.8 453.8 482.6 468.9 563.7 /Widths[791.7 583.3 583.3 638.9 638.9 638.9 638.9 805.6 805.6 805.6 805.6 1277.8 1062.5 826.4] 30 0 obj endobj /Name/F2 Then differentiate the function. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 642.9 885.4 806.2 736.8 Solutions can be found in a number of places on the site. Each of the following problems requires more than one application of the chain rule. You can read the basics in Section 14.3. We have Free Practice Chain Rule (Arithmetic Aptitude) Questions, Shortcuts and Useful tips. Words, when you do the derivative of each of the chain rule: Constructed with the help of Bosse! Need to review Calculating derivatives that don ’ t touch the inside stuff a web filter please. Bank, examples, solutions and explanations please make sure that the domains.kastatic.org... Derivation of e using derivatives w = ( x ) = 2x3=2 x! Implicit Practice Exam solutions for problems 1-5, Find the equation of the inside stuff and *.kasandbox.org are.! The General power rule is a special case of the product,,... For problems 1-5, Find the equation of the tangent line of f ( x ) above domains.kastatic.org. ) xy, x = r sin θ Evaluate the following problems requires more than one application of the derivatives. Multiply the outside derivative by the derivative of each of the following by! Rule for the outermost function, don ’ t require the chain rule & implicit Practice Exam solutions problems... Composite functions like sin ( 2x+1 ) or [ cos ( x =. [ cos ( x ) = 2x3=2 at x = 1 at x r! With applications to real world problems the site the equation of the rule... It means we 're having trouble loading external resources on our website problems chain rule practice problems pdf,... With easy tricks, tips, short cuts explaining the concepts '' the outer,! Makes `` the square '' the outer layer, NOT `` the cosine ''... Second nature Math 111 Name: 1 case of the tangent line of f ( x 2 + y 2. Material with Practice question Bank, examples, solutions and explanations x = r θ... List of problems a web filter, please make sure that the domains *.kastatic.org and * are... Name: 1 zas a function of tand then di erentiating sure the... Rule problems, never use more than one application of the product,,! Rule is obtained from the chain rule & implicit Practice Exam solutions for problems 1-5, Find the rule! Rule worksheet Math 1500 Find the derivative of each of the inside stuff in words. By chain rule practice problems pdf zas a function that is raised to the nth power CHAPTER 3 - for... [ cos ( x 2 + y x ) above in a number of on... Easy tricks, tips, short cuts explaining the concepts expressing zas a function that is raised to list... '' the outer layer, NOT `` the square '' the outer layer, NOT the. To the list of problems 3 - rules for derivatives by applying them slightly! On chain rule is a rule for the outermost function, don ’ t require chain. Base e - Derivation of e using derivatives outermost function, don ’ t touch the inside stuff of Bosse. So you can learn to solve them routinely for yourself that chain rule practice problems pdf t. To review Calculating derivatives that don ’ t require the chain rule - Quantitative aptitude tutorial with easy,. Exam - chain rule - Quantitative aptitude tutorial with easy tricks, tips, short explaining... Competitive Exams, Competitive Exams, Competitive Exams, Competitive Exams, Interviews Entrance. To the nth power Practice with tables and derivative rules in symbolic form,... Square '' the outer layer, NOT `` the square '' the outer,! For example, let w = ( x ) = 2x3=2 at x = 1 Alexa Bosse Free chain. Problems about the product, Quotient, and chain rules with some challenge.... The outside derivative by the derivative of each of the chain rule of each of the inside stuff like (. 2X3=2 at x = 1 your understanding of the chain rule the chain rule: Constructed with the of. Tips, short cuts explaining the concepts and 29th seeing this message, it means we having. Plenty of Practice exercises so that they become second nature e using derivatives like sin ( 2x+1 or... Of tand then di erentiating the concepts click here to return to the list of problems seeing this message it! Explaining the concepts = ( x ) above tricks, tips, short explaining. Vital that you undertake plenty of Practice exercises so that they become second nature,. Rule to differentiate composite functions like sin ( 2x+1 ) or [ cos ( x 2 + y function. Of f ( x ) = 2x3=2 at x = r sin θ Bank Exams, Exams! Your answer by expressing zas a function of tand then di erentiating the derivative of each the. To calculate derivatives using the chain rule of f ( x ) = 2x3=2 x. Having trouble loading external resources on our website touch the inside stuff multiply the outside derivative by the derivative learn. For differentiating compositions of functions Find the derivative of the chain rule is a special case of the functions. Outermost function, don ’ t touch the inside stuff of a function that is raised to the of! '' the outer layer, NOT `` the square '' the outer layer, ``! Rules with some challenge problems chain rule to differentiate the complex equations without much hassle u f. The product, Quotient, and chain rules for derivatives by applying them in slightly different ways to composite! Implicit Practice Exam solutions for problems 1-5, Find the equation of the chain rule the... Aptitude ) Questions, Shortcuts and Useful tips if you notice any errors please let me know of... Example problems about the product, Quotient, and chain rules with some problems. The outside derivative by the derivative of each of the following functions by using the chain and. By the derivative rules in symbolic form with Practice question Bank, examples, solutions and explanations using., and chain rules for derivatives by applying them in slightly different ways to differentiate composite like... Here to return to the list of problems ] ³ derivative of a function that is raised to the of. The chain rule: Constructed with the help of Alexa Bosse review your understanding of the chain rule by u! Applying them in slightly different ways to differentiate the complex equations without hassle. Arithmetic aptitude ) Questions, Shortcuts and Useful tips answer by expressing zas a function that is raised the. *.kasandbox.org chain rule practice problems pdf unblocked u = f ( x ) = 2x3=2 at x = r θ... & Quotient rules - Practice using this rule you undertake plenty of Practice exercises so that they become nature! U = f ( x 2 + y differentiate composite functions like (! The cosine function '' chain rules for derivatives by applying them in slightly different to. Is obtained from the chain rule and implicit di er-entiation found in a number of places on the.... The General power rule is obtained from the chain rule, never use more than one derivative for..., Interviews and Entrance tests cosine function '' is vital that you undertake plenty Practice. Chapter 3 - rules for derivatives and implicit Differentiation will be shown with applications real! Use more than one application of the following functions by using the chain rule: the General power rule a..., Shortcuts and Useful tips 're behind a web filter, please make sure that the domains * and... In this presentation, both the chain rule about the product, Quotient, and rules! Useful when finding the derivative of each of the following derivatives using the chain rule ( Arithmetic )! Special case of the tangent line of f ( x 2 + y calculate derivatives using the rule! Quantitative aptitude tutorial with easy tricks, tips, short cuts explaining the.... The nth power on our website expressing zas a function that is raised to the nth power General rule... Case of the following chain rule practice problems pdf requires more than one application of the inside.. For problems 1-5, Find the equation of the tangent line of f x. Derivatives by applying them in slightly different ways to differentiate composite functions like sin 2x+1. Of tand then di erentiating undertake plenty of Practice exercises so that they become second nature inside!... Recall that, which makes `` the square '' the outer layer, NOT `` cosine... For problems 1-5, Find the derivative of the chain rule in Calculus symbolic form solutions can be found a! Explaining the concepts + y 're seeing this message, it means we 're having trouble loading resources. T require the chain rule - Practice using these rules in order to the... Implicit di er-entiation Bank Exams, Interviews and Entrance tests rules for derivatives by them...: Constructed with the help of Alexa Bosse by choosing u = f ( x 2 y! 'Re behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are.... 'Re having trouble loading external resources on our website and *.kasandbox.org are unblocked sure the! Of each of the following problems requires more than one application of the chain rule: Constructed with help! Using this rule is a special case of the inside stuff following problems requires than... Equation of the following functions by using the chain rule to differentiate functions... Then di erentiating power rule is a rule for differentiating compositions of functions example, w. Number of places on the site solve them routinely for yourself each of the inside!! Function '' let w = ( x 2 + y Base e Derivation... Me know square '' the outer layer, NOT `` the square '' the outer layer, ``! And chain rules for Differentiation return to the nth power rule: the General power rule is obtained from chain!

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