[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-522814-en":3,"detail-sidebar-cat-0-en-105":30,"doc-seo-522814-105":80},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":11,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29,"preview_page_count":4},522814,1099523885074,"Riley West","https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc",4,"Exam","Differentiation - Chain Rule - Worksheet 03","Differentiation worksheet on the chain rule from Infinite Calculus, asking students to differentiate a set of polynomial and composite functions with respect to x. It includes problems that require applying repeated chain rule steps, including products of functions and nested exponents. A guided solution section demonstrates derivative setup and simplification for several items. The worksheet also contains a critical thinking prompt where students construct a function requiring three chain rule applications, then compute its derivative.","Kuta Software-Infinite Calculus Name   \nDifferentiation-Chain Rule Date  Period   \nDifferentiate each function with respect to x.  \n1) y = (x3 + 3) 5 2) y = (−3x5 + 1) 3  \n3) y = (−5x3 − 3) 3 4) y = (5x2 + 3)4  \n5) f (x) = 4 −3x4 − 2 6) f (x) = −2x2 + 1  \n7) f (x) = −2x 8) y = (−x4 − 3)−2  \n9) y = (3x3 + 1)(−4x2 − 3)4  \n(x3 + 4) 5   \n10) y =  \n3x4 − 2  \n11) y = ((x + 5) 5 − 1)4  \n12) y = (5x3 − 3) 5 4 −4x5 − 3   \nCritical thinking question:  \n13) Give a function that requires three applications of the chain rule to differentiate. Then differentiate the function.  \nKuta Software-Infinite Calculus Name   \nDifferentiation-Chain Rule Date  Period   \nDifferentiate each function with respect to x.  \n1) y = (x3 + 3) 5  \ndydx = 5(x3 + 3)4 ⋅ 3x2 = 15x2 (x3 + 3)4  \n3) y = (−5x3 − 3) 3  \ndydx = 3(−5x3 − 3)2 ⋅ −15x2 = −45x2 (−5x3 − 3)2  \n5) f (x) = 4 −3x4 − 2   \n1 −  3  \nf ' (x) =  4  (−3x4 − 2) 4 ⋅ −12x3  3x3   \n= −  \n3  \n(−3x4 − 2) 4  \n7) f (x) = 3 −2x4 + 5   \n1 −  2 f ' (x) =  3  (−2x4 + 5) 3 ⋅ −8x3  \n 8x3 = −  \n2  \n3(−2x4 + 5) 3  \n2) y = (−3x5 + 1) 3  \ndydx = 3(−3x5 + 1)2 ⋅ −15x4 = −45x4 (−3x5 + 1)2  \n4) y = (5x2 + 3)4  \ndydx = 4(5x2 + 3) 3 ⋅ 10x  \n= 40x(5x2 + 3) 3  \n6) f (x) = −2x2 + 1  \n1  \nf ' (x) = ~~ ~~12~~ ~~ (−2x2 + 1) −  2 ⋅ −4x  \n 2x = −  \n1  \n(−2x2 + 1) 2  \n8) y = (−x4 − 3)−2  \ndydx = −2(−x4 − 3)−3 ⋅ −4x3  \n 8x3 =  \n(−x4 − 3) 3  \n9) y = (3x3 + 1)(−4x2 − 3)4  \ndydx = (3x3 + 1) ⋅ 4(−4x2 − 3) 3 ⋅ −8x + (−4x2 − 3)4 ⋅ 9x2 = x(−4x2 − 3) 3 (−132x3 − 32 − 27x)  \n(x3 + 4) 5   \n10) y = 4  \n3x − 2  \ndy (3x4 − 2) ⋅ 5(x3 + 4) 4 ⋅ 3x2 − (x3 + 4) 5 ⋅ 12x3 ~~ ~~ = ~~  ~~dx (3x4 − 2)2  \n=  \n3x2 (x3 + 4) 4 ( 11x4 − 10 − 16x)  \n(3x4 − 2)2  \n11) y = ((x + 5) 5 − 1)4  \ndydx = 4((x + 5) 5 − 1) 3 ⋅ 5(x + 5)4  \n= 20((x + 5) 5 − 1) 3 ⋅ (x + 5)4  \n12) y = (5x3 − 3) 5 4 −4x5 − 3   \n3 1  \ndydx = (5x3 − 3) 5 ⋅ ~~ ~~14~~ ~~ (−4x5 − 3) −  4 ⋅ −20x4 + (−4x5 − 3)  4 ⋅ 5(5x3 − 3)4 ⋅ 15x2  \n5x2 (5x3 − 3) 4 (−65x5 − 45 + 3x2 )   \n=  \n3  \n(−4x5 − 3) 4  \nCritical thinking question:  \n13) Give a function that requires three applications of the chain rule to differentiate. Then differentiate the function.  \nMany answers: Ex y = (((2x + 1) 5 + 2) 6 + 3)7  \ndydx = 7(((2x + 1) 5 + 2) 6 + 3) 6 ⋅ 6((2x + 1) 5 + 2) 5 ⋅ 5(2x + 1)4 ⋅ 2  \nCreate your own worksheets like this one with Infinite Calculus. Free trial available [at KutaSoftware.com](at KutaSoftware.com)","cbCaidGwgcb7ddxM","https://ap.wps.com/l/cbCaidGwgcb7ddxM","pdf",38632,11,1,"English","en",105,"# Differentiate each function with respect to x\n## Critical thinking question","[{\"question\":\"What does the worksheet ask students to do for each problem?\",\"answer\":\"Differentiate each given function with respect to x, using the chain rule where appropriate.\"},{\"question\":\"Does the worksheet include a critical thinking question?\",\"answer\":\"Yes. Students must give a function that requires three applications of the chain rule and then differentiate it.\"},{\"question\":\"What kinds of functions appear in the differentiation problems?\",\"answer\":\"The worksheet includes composite expressions (nested powers) and at least one product of functions that must be differentiated using chain rule steps.\"}]","Differentiation - Chain Rule - Worksheet 03 | PDF",1791426201,10,{"code":4,"msg":5,"data":31},[32,37,42,45,50,55,60,65,70,73,76],{"id":21,"doc_module":4,"doc_module_name":33,"category_name":34,"show_sort_weight":35,"slug":36},"Document","Story & Novel",90,"story-novel",{"id":38,"doc_module":4,"doc_module_name":33,"category_name":39,"show_sort_weight":40,"slug":41},2,"Literature",80,"literature",{"id":11,"doc_module":4,"doc_module_name":33,"category_name":12,"show_sort_weight":43,"slug":44},70,"exam",{"id":46,"doc_module":4,"doc_module_name":33,"category_name":47,"show_sort_weight":48,"slug":49},5,"Comic",60,"comic",{"id":51,"doc_module":4,"doc_module_name":33,"category_name":52,"show_sort_weight":53,"slug":54},6,"Technology",50,"technology",{"id":56,"doc_module":4,"doc_module_name":33,"category_name":57,"show_sort_weight":58,"slug":59},7,"Healthcare",40,"healthcare",{"id":61,"doc_module":4,"doc_module_name":33,"category_name":62,"show_sort_weight":63,"slug":64},8,"Research & Report",30,"research-report",{"id":66,"doc_module":4,"doc_module_name":33,"category_name":67,"show_sort_weight":68,"slug":69},9,"Religion & Spirituality",20,"religion-spirituality",{"id":68,"doc_module":4,"doc_module_name":33,"category_name":71,"show_sort_weight":68,"slug":72},"World Cup","world-cup",{"id":29,"doc_module":4,"doc_module_name":33,"category_name":74,"show_sort_weight":29,"slug":75},"Lifestyle","lifestyle",{"id":77,"doc_module":4,"doc_module_name":33,"category_name":78,"show_sort_weight":46,"slug":79},19,"General","general",{"code":4,"msg":81,"data":82},"ok",{"site_id":24,"language":23,"slug":83,"title":13,"keywords":84,"description":14,"schema_data":85,"social_meta":139,"head_meta":141,"extra_data":143,"updated_unix":144},"differentiation-chain-rule-worksheet-03","",{"@graph":86,"@context":138},[87,101,121],{"@type":88,"itemListElement":89},"BreadcrumbList",[90,94,96,99],{"item":91,"name":92,"@type":93,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":95,"name":33,"@type":93,"position":38},"https://docshare.wps.com/document/",{"item":97,"name":12,"@type":93,"position":98},"https://docshare.wps.com/document/exam/",3,{"item":100,"name":13,"@type":93,"position":11},"https://docshare.wps.com/document/differentiation-chain-rule-worksheet-03/522814/",{"url":100,"name":13,"@type":102,"image":103,"author":108,"headline":13,"publisher":110,"fileFormat":113,"inLanguage":23,"description":14,"dateModified":114,"datePublished":115,"encodingFormat":113,"isAccessibleForFree":116,"interactionStatistic":117},"DigitalDocument",{"url":104,"@type":105,"width":106,"height":107},"https://docshare.wps.com/thumbnails/differentiation-chain-rule-worksheet-03/522814.png","ImageObject",300,407,{"name":9,"@type":109},"Person",{"url":91,"name":111,"@type":112},"DocShare","Organization","application/pdf","2026-11-10","2026-10-08",true,{"@type":118,"interactionType":119,"userInteractionCount":20},"InteractionCounter",{"@type":120},"ViewAction",{"@type":122,"mainEntity":123},"FAQPage",[124,130,134],{"name":125,"@type":126,"acceptedAnswer":127},"What does the worksheet ask students to do for each problem?","Question",{"text":128,"@type":129},"Differentiate each given function with respect to x, using the chain rule where appropriate.","Answer",{"name":131,"@type":126,"acceptedAnswer":132},"Does the worksheet include a critical thinking question?",{"text":133,"@type":129},"Yes. Students must give a function that requires three applications of the chain rule and then differentiate it.",{"name":135,"@type":126,"acceptedAnswer":136},"What kinds of functions appear in the differentiation problems?",{"text":137,"@type":129},"The worksheet includes composite expressions (nested powers) and at least one product of functions that must be differentiated using chain rule steps.","https://schema.org",{"og:url":100,"og:type":140,"og:title":13,"og:site_name":111,"og:description":14},"article",{"robots":142,"canonical":100},"index,follow",{"doc_id":7,"site_id":24},1791549769]