[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-522841-105":59,"doc-detail-522841-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","differentiation-product-rule-worksheet-3","Differentiation - Product Rule - Worksheet 3","","Differentiation worksheets on the product rule for derivatives, asking students to differentiate multiple algebraic functions with respect to x. The document lists problems in a numbered sequence and provides detailed worked derivatives for each function, including cases with polynomials, negative coefficients, and products of expressions. It also includes a critical-thinking item challenging a claim about derivatives of a product and requiring an example to refute it, with one sample choice of functions.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/differentiation-product-rule-worksheet-3/522841/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/differentiation-product-rule-worksheet-3/522841.png","ImageObject",300,407,{"name":92,"@type":93},"Taylor Morgan","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-12","2026-10-08",true,{"@type":102,"interactionType":103,"userInteractionCount":19},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What rule is used to differentiate products of functions in these problems?","Question",{"text":112,"@type":113},"The worksheet focuses on the product rule for derivatives, differentiating a product of two functions with respect to x.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"Does the worksheet include answers or only questions?",{"text":117,"@type":113},"It includes worked derivatives for the listed differentiation problems, showing the derivative steps.",{"name":119,"@type":110,"acceptedAnswer":120},"How can you refute the claim about (f·g)' equaling f'·g'?",{"text":121,"@type":113},"You must provide an example of functions f and g where (f·g)' differs from f'·g'. The worksheet includes a critical-thinking question and gives an example idea (e.g., f = 2x, g = 4) as a starting point.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},522841,1791536326,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":19,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":19,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":143,"read_time":52},1099523885336,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","Kuta Software-Infinite Calculus Name___________________________________  \nDifferentiation-Product Rule Date________________ Period____  \nDifferentiate each function with respect to x.  \n1) y = −x3 (3x4 − 2) 2) f (x) = x2 (−3x2 − 2)  \n3) y = (−2x4 − 3)(−2x2 + 1) 4) f (x) = (2x4 − 3)(x2 + 1)  \n5) f (x) = (5x5 + 5)(−2x5 − 3) 6) f (x) = (−3 + x−3 )(−4x3 + 3)  \n7) y = (−2x4 + 5x2 + 4)(−3x2 + 2)  \n8) y = ( x4 + 3)(−4x5 + 5x4 + 5)  \n9) y = (5x4 − 3x2 − 1)(−5x2 + 3)  \n10) f (x) = (−10x2 − 7 5 x2 + 9)(2x3 + 4)  \n11) y = (5 + 3x−2 )(4x5 + 6x3 + 10)  \n12) y = (−6x4 + 2 + 6x−4 )(6x4 + 7)  \n13) f (x) = (−7x4 + 10x25~~ ~~ + 8)(x2 + 10)  \nCritical thinking question:  \n14) A classmate claims that ( f ⋅ g) ' = f ' ⋅ g ' for any functions f and g. Show an example that proves your classmate wrong.  \nKuta Software-Infinite Calculus  \nDifferentiation-Product Rule  \nDifferentiate each function with respect to x.  \n1) y = −x3 (3x4 − 2)  \ndydx = −x3 ⋅ 12x3 + (3x4 − 2) ⋅ −3x2 = −21x6 + 6x2  \nName___________________________________  \nDate________________ Period____  \n2) f (x) = x2 (−3x2 − 2)  \nf ' (x) = x2 ⋅ −6x + (−3x2 − 2) ⋅ 2x  \n= −12x3 − 4x  \n3) y = (−2x4 − 3)(−2x2 + 1)  \ndydx = (−2x4 − 3) ⋅ −4x + (−2x2 + 1) ⋅ −8x3 = 24x5 − 8x3 + 12x  \n5) f (x) = (5x5 + 5)(−2x5 − 3)  \nf ' (x) = (5x5 + 5) ⋅ −10x4 + (−2x5 − 3) ⋅ 25x4 = −100x9 − 125x4  \n4) f (x) = (2x4 − 3)(x2 + 1)  \nf ' (x) = (2x4 − 3) ⋅ 2x + (x2 + 1) ⋅ 8x3 = 12x5 + 8x3 − 6x  \n6) f (x) = (−3 + x−3 )(−4x3 + 3)  \nf ' (x) = (−3 + x−3 ) ⋅ −12x2 + (−4x3 + 3) ⋅ −3x−4 = 36x2 − 9x4~~ ~~  \n7) y = (−2x4 + 5x2 + 4)(−3x2 + 2)  \ndydx = (−2x4 + 5x2 + 4) ⋅ −6x + (−3x2 + 2)(−8x3 + 10x)= 36x5 − 76x3 − 4x  \n8) y = ( x4 + 3)(−4x5 + 5x4 + 5)  \ndydx = (x4 + 3)(−20x4 + 20x3 ) + (−4x5 + 5x4 + 5) ⋅ 4x3 = −36x8 + 40x7 − 60x4 + 80x3  \n9) y = (5x4 − 3x2 − 1)(−5x2 + 3)  \ndydx = (5x4 − 3x2 − 1) ⋅ −10x + (−5x2 + 3)(20x3 − 6x)= −150x5 + 120x3 − 8x  \n10) f (x) = (−10x2 − 7 5 x2 + 9)(2x3 + 4)  \nf ' (x) = (−10x2 − 7x25 + 9) ⋅ 6x2 + (2x3 + 4)(−20x − ~~ ~~145x − ~~ ~~35~~ ~~ )  \n 12   \n5  \n= −100x4 − 238x5~~ ~~ + 54x2 − 80x − ~~ ~~563~~  ~~  \n5  \n5x  \n11) y = (5 + 3x−2 )(4x5 + 6x3 + 10)  \ndydx = (5 + 3x−2 )(20x4 + 18x2 ) + (4x5 + 6x3 + 10) ⋅ −6x−3 = 100x4 + 126x2 + 18 − ~~ ~~6x03  \n12) y = (−6x4 + 2 + 6x−4 )(6x4 + 7)  \ndydx = (−6x4 + 2 + 6x−4 ) ⋅ 24x3 + (6x4 + 7)(−24x3 − 24x−5 )= −288x7 − 120x3 − ~~ ~~16x58  \n13) f (x) = (−7x4 + 10x25~~ ~~ + 8)(x2 + 10)  \nf ' (x) = (−7x4 + 10x25 + 8) ⋅ 2x  + (x2 + 10)(−28x3 + 4x − ~~ ~~35~~ ~~ )  \n7  \n= −42x5 − 280x3 + 24x 5 + 16x + ~~ ~~403  \n5  \nx  \nCritical thinking question:  \n14) A classmate claims that ( f ⋅ g) ' = f ' ⋅ g ' for any functions f and g. Show an example that proves your classmate wrong.  \nMany answers. Ex: f = 2x, g = 4, 8 ≠ 0  \nCreate your own worksheets like this one with Infinite Calculus. Free trial available [at KutaSoftware.com](at KutaSoftware.com)","cbCaigzJxoILEUEU","https://ap.wps.com/l/cbCaigzJxoILEUEU","pdf",41081,"English","# Differentiation - Product Rule\n## Practice problems (1-13)\n## Critical thinking question (14)","[{\"question\":\"What rule is used to differentiate products of functions in these problems?\",\"answer\":\"The worksheet focuses on the product rule for derivatives, differentiating a product of two functions with respect to x.\"},{\"question\":\"Does the worksheet include answers or only questions?\",\"answer\":\"It includes worked derivatives for the listed differentiation problems, showing the derivative steps.\"},{\"question\":\"How can you refute the claim about (f·g)' equaling f'·g'?\",\"answer\":\"You must provide an example of functions f and g where (f·g)' differs from f'·g'. The worksheet includes a critical-thinking question and gives an example idea (e.g., f = 2x, g = 4) as a starting point.\"}]","Differentiation - Product Rule - Worksheet 3 | PDF",1791426225]