[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-459832-105":59,"doc-detail-459832-en":131},{"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":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","unit-3-mcq-ap-calculus-practice-test-solutions","Unit 3 MCQ - AP Calculus Practice Test Solutions","","AP Calculus practice test solutions for Unit 3 MCQ, presenting step-by-step answers to multiple derivative problems. The document emphasizes using the chain rule, including differentiating compositions and implicit functions. It also contrasts when the chain rule is appropriate versus when the product rule should be used, and applies inverse function differentiation and inverse trig differentiation. The questions cover tangent line slopes, inverse sine and inverse cosine derivatives, and selecting correct differentiation methods.",{"@graph":69,"@context":123},[70,84,106],{"@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/unit-3-mcq-ap-calculus-practice-test-solutions/459832/",{"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/unit-3-mcq-ap-calculus-practice-test-solutions/459832.png","ImageObject",300,407,{"name":92,"@type":93},"CatatanPagi","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-27","2026-09-30",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",22,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"How does the chain rule apply to derivatives of composite functions in this document?","Question",{"text":113,"@type":114},"The solutions state that if an expression is a composition, the chain rule differentiates the outside function and multiplies by the derivative of the inside function.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"When should implicit differentiation be used instead of the chain rule?",{"text":118,"@type":114},"The solutions explain that for equations where the dependent variable cannot be isolated, implicit differentiation is required, such as slope problems from implicit curves.",{"name":120,"@type":111,"acceptedAnswer":121},"How are inverse functions and inverse trigonometric functions differentiated here?",{"text":122,"@type":114},"The solutions use inverse-function relationships (from f(g(z))=z) and also differentiate inverse sine/cosine by converting to an implicit trig equation, then applying the chain rule and selecting the correct square-root sign.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},459832,1790813783,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":140,"language":141,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":67,"update_tm":145,"read_time":146},962090894170,"https://ap-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8","AP Calculus Practice test solutions  \nUnit 3 MCQ  \n# ⑬⑭516⑰18Question 1\n\nIf g(z)=lnz and fis a differentiable function of z,which of the following is equivalent to the derivative of f(g(z)with respect to z?  \n④f'()①Answer D  \nCorect.The chain rule provides a way to diferentate a composion oftuntions.The“outside”funtionis f(z)and the “inside”funtionis s(z)=1mz:Theng(z)=an the chainrule can be used,as follows.  \nAP Calculus Practice test solutions  \n①②  \n⑰1⑱  \n2 of 18  \n## Question 2\n\nFor which of the following functions is the chain rule an appropriate method to find the derivative with respect to z?  \n1.y=sin(3z²)  \nI.y=etan zAI only⑥◎IandIⅢl only  \nAnswer D  \nCorrectEach of the functions in l and Ⅲ is a composition of component functions.  \nIn I,let f(z)=sin z and let g(z)=3z².Then sin(3z²)=f(9(z).and the chain rule would give sin(3z²)=f'(g(z)s(z)=cos(3z²)6z.),and the chain rule would give  \nn ll,the function y=e²tan z is the product of the exponential function and the tangent function and is not the composition of two component functions.The product rule would be usedtofind the derivative,and the chain rule would not be appropriate.  \nAP Calculus Practice test solutions  \n## Question 3\n\nLet fbe a differentiable function.If h(x)=(1+f(3x))²,which of the following gives a correct process for finding h'(z)?  \nA  \nh'(x)=2(1+f(3x))  \n⑧  \nh'(z)=2(1+f(3x))·f'(3x)  \nCh'(z)=2(1+f(3x)·f'(z)  \n①  \nh'(x)=2(1+f(3x))·f'(3x)·3  \nAnswer D  \nCorrect.This requires using the chain rule twice to compute the derivative.  \nAP Calculus Practice test solutions  \n## Question 4\n\nWhat is the slope of the line tangent to the curve y³-xy²+z³=5at the point (1,2)?  \n⑧  \n8  \n①  \nAnswer B  \nCorrect.The point (1,2)is on the curve since z=1,y=2 sattisfies the implicit equation ofthe curve.The chain rule is the basis for implicit differentiation.+3z²=0At the point (1,2),1  \nAP Calculus Practice test solutions  \nQuestion 5  \nA  \n⑧  \nC  \n0m621场)  \n①  \nAnswer D  \nCorrect.The chain rule is the basis for implicit differentiation as well as the differentiation of a composite function.  \nAP Calculus Practice test solutions  \n①②③  \n⑪⑰8  \n6 of 18  \nf'(-2)=4  \ng(4)=5  \nf(-2)=3  \ng'(4)=2  \nThe point(-2,4)lies on thecure in the zyplane given byte equation f(z)9(y)=17一T-y.where fisadfifentable functionof z and gisa diferentabl function of y selectedvalues of f.f',g.and g'are given in the table above.What is the value(  \n⑧  \n①  一  \nAnswer C  \nAP Calculus Practice test solutions  \n5 ⑯⑪⑱  \nQuestion 7  \nBockr  \nThegraph of the decreasing dfferentlable function fis shown above.Also shown is the line tangent to the graph of fat the point(4,6).Let gbe the inverse of f.Which of the following statements about g'is true?  \nA g(4)=-  \n⑧g(4)=-  \n◎ g(6)=-4  \n①g(6)=-  \nAnswer c  \nCorrect This statementcan be confrmed using the chain nule and the definifion of an inwerse function.The line tangent to the graph of fat the point(4,6)has slope f(4)=-1.Since g(z)=F'(z),then g(6)=4and f(q(z)=z By the chain rule,  \n①②  \n8of 18  \n## Question 8\n\nLet fbe the increasing function defne by f(z)=z³+2z²+4r+5,where f(-1)=2.fgisthe inverse function of f,which ofthe fllowing isa coret expression fo g'(2)?  \nA  \n Bookmark  \n⑧  \nC g'(2)=f'(-1)  \n①  g'(2)=f'(2)  \nAnswer B  \nCorrect.This can be confirmed using the chain rule and the definition of an inverse function.Since f(g(z))=z,it follows thatTherefore,  \nAP Calculus Practice test solutions  \n①②③④⑤⑥⑦⑧◎ 1⑩9 of 18  \n\n| ①   | 0   | 2   | 4   |\n| --- | --- | --- | --- |\n| f(z)   | 8   | 5   | 2   |\n| f'(z)   | -1   | -2   | -5   |\n\nThetable above gies selected aluesfora diferentiableand decreasing function fand ts derivatve.f (z)=f¹(z)fral,whic of te folowing isa coret expresion for g(2)?  \nA g'(2)=f'(2)=-2  \n⑧  \n@  \n①  \nAnswer C  \nCorrect.Ths value can be confimed using the chain rule and the definition of an inverse function.Since f(g(z))=z,it follows that  \n10 of 18  \nQuestion 10  \nA  \n⑥  \n◎ cos¹()  \n①  -csc(2)cot()  \nAnswer B  \nCorrect.If y=sin⁻¹z,then sin","cbCaimholcBQiU8Q","https://ap.wps.com/l/cbCaimholcBQiU8Q","pdf",682751,14,"English","# Question 1\n# Question 2\n# Question 3\n# Question 4\n# Question 5\n# Question 6\n# Question 7\n# Question 8\n# Question 9\n# Question 10\n# Question 11\n# Question 12\n# Question 13\n# Question 14","[{\"question\":\"How does the chain rule apply to derivatives of composite functions in this document?\",\"answer\":\"The solutions state that if an expression is a composition, the chain rule differentiates the outside function and multiplies by the derivative of the inside function.\"},{\"question\":\"When should implicit differentiation be used instead of the chain rule?\",\"answer\":\"The solutions explain that for equations where the dependent variable cannot be isolated, implicit differentiation is required, such as slope problems from implicit curves.\"},{\"question\":\"How are inverse functions and inverse trigonometric functions differentiated here?\",\"answer\":\"The solutions use inverse-function relationships (from f(g(z))=z) and also differentiate inverse sine/cosine by converting to an implicit trig equation, then applying the chain rule and selecting the correct square-root sign.\"}]","Unit 3 MCQ - AP Calculus Practice Test Solutions | PDF",1790757043,35]