[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-669658-105":59,"doc-detail-669658-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","derivative-formulas-derivative-rules-and-applications","DERIVATIVE FORMULAS - derivative rules and applications","","Derivative formulas for constant, sum, difference, product, quotient, and power rules, including versions with and without the chain rule. Covers chain rule structures and derivative patterns for exponential functions, natural logarithms, and general logarithms. Includes absolute value differentiation guidance and trigonometric derivatives (sin, csc, cos, sec, tan, cot) with their chain-rule forms. Also provides inverse trigonometric derivative expressions involving radicals such as sqrt(1-x^2) and related terms.",{"@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":30,"@type":76,"position":81},"https://docshare.wps.com/document/technology/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/derivative-formulas-derivative-rules-and-applications/669658/",{"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/derivative-formulas-derivative-rules-and-applications/669658.png","ImageObject",300,407,{"name":92,"@type":93},"Taylor Morgan","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-02","2026-11-01",true,{"@type":102,"interactionType":103,"userInteractionCount":8},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What derivative rules are included besides the power rule?","Question",{"text":112,"@type":113},"The document lists constant rule, basic sum rule, difference rule, product rule, and quotient rule, in addition to the power rule and chain rule structures.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How does the chain rule affect the derivative formulas shown?",{"text":117,"@type":113},"It provides corresponding derivative versions that multiply by the derivative of the inside function, compared to formulas without chain rule.",{"name":119,"@type":110,"acceptedAnswer":120},"What function types are covered for special formulas?",{"text":121,"@type":113},"It includes exponential functions, natural logarithms and other logarithms, absolute value, trigonometric functions, and inverse trigonometric functions.","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},669658,1793563441,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":29,"category_name":30,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":8,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":8,"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":129,"read_time":81},1099523885336,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","DERIVATIVE FORMULAS  \nConstant Rule Basic Sum Rule Difference Rule Product Rule Quotient Rule  \n􀝀􀝀􀝔 [􀜿] = 0 ~~ ~~􀝀􀝀􀝔 [􀝔] = 1 ~~ ~~􀝀􀝀􀝔 [􀝑 + 􀝒] = 􀝑 ′ + 􀝒′ ~~ ~~􀝀􀝀􀝔 [􀝑 − 􀝒] = 􀝑 ′ − 􀝒′ ~~ ~~􀝀􀝀􀝔 [􀝑􀝒] = 􀝑 ′􀝒 + 􀝑􀝒′ ~~ ~~􀝀􀝀􀝔 [􀝑􀝒] = 􀝑~~ ~~′􀝒~~ ~~~~ ~~􀝑􀝒′  \nWITHOUT CHAIN RULE  WITH CHAIN RULE  \nPower Rule Chain Rule  \n􀝀􀝀􀝔 [􀝔􀯡 ] = 􀝊􀝔􀯡−1 ~~ ~~􀝀􀝀􀝔 [􀝑􀯡 ] = 􀝊􀝑􀯡−1 􀝑 ′  \n􀝀􀝀􀝔 [􀜿􀝔􀯡 ] = 􀜿􀝊􀝔􀯡−1  ~~ ~~􀝀􀝀􀝔 [􀜿􀝑􀯡 ] = 􀜿􀝊􀝑􀯡−1 􀝑 ′  \nExponential Exponential  \n􀝀􀝀􀝔 [􀝁 􀯫 ] = 􀝁 􀯫 ~~ ~~􀝀􀝀􀝔 [􀝁 􀯨 ] = 􀝁 􀯨 􀝑 ′  \nNatural Log Natural Log  \n􀝀􀝀􀝔 [ln 􀝔] = 1􀝔 ~~ ~~􀝀􀝀􀝔 [ln 􀝑] = 1􀝑 􀝑 ′  \nLogarithmic Logarithmic  \n􀝀􀝀􀝔 [log􀯔 􀝔] = ln1􀜽 ∙ 1􀝔 ~~ ~~􀝀􀝀􀝔 [log􀯔 􀝑] = ln1􀜽 ∙ 1􀝑 ∙ 􀝑 ′  \nAbsolute Value Absolute Value  \n􀝀􀝀􀝔 [|􀝔|] = |􀝔􀝔~~ ~~| ~~ ~~􀝀􀝀􀝔 [|􀝑|] = ~~ ~~|􀝑􀝑~~ ~~| 􀝑 ′  \nTrig Functions  Trig Functions  \n􀝀􀝀􀝔 [sin 􀝔] = cos 􀝔 ~~ ~~􀝀􀝀􀝔 [csc 􀝔] = −csc 􀝔 cot 􀝔 ~~ ~~􀝀􀝀􀝔 [sin 􀝑] = (cos 􀝑)􀝑 ′ ~~ ~~􀝀􀝀􀝔 [csc 􀝑] = −(csc 􀝑 cot 􀝑)􀝑 ′􀝀􀝀􀝔 [cos 􀝔] = −sin 􀝔 ~~ ~~􀝀􀝀􀝔 [sec 􀝔] = sec 􀝔 tan 􀝔 ~~ ~~􀝀􀝀􀝔 [cos 􀝑] = −(sin 􀝑)􀝑 ′ ~~ ~~􀝀􀝀􀝔 [sec 􀝑] = (sec 􀝑 tan 􀝑)􀝑 ′􀝀􀝀􀝔 [tan 􀝔] = sec 2 􀝔 ~~ ~~􀝀􀝀􀝔 [cot 􀝔] = − csc 2 􀝔 ~~ ~~􀝀􀝀􀝔 [tan 􀝑] = (sec2 􀝑) 􀝑 ′ ~~ ~~􀝀􀝀􀝔 [cot 􀝑] = −(csc2 􀝑)􀝑 ′  \nInverse Trig Functions  Inverse Trig Functions  \n 􀝀  [sin−1 􀝔] =  1   􀝀  [csc−1 􀝔] =  −1   􀝀  [sin−1 􀝑] =  􀝑 ′   􀝀  [csc−1 􀝑] =  −􀝑′ 􀝀􀝔 √1 − 􀝔 2 􀝀􀝔 |􀝔|√􀝔2 − 1 􀝀􀝔 √1 − 􀝑 2 􀝀􀝔 |􀝑|√􀝑2 − 1  \n 􀝀  −1  􀝀  1  􀝀  −􀝑′  􀝀  􀝑 ′[cos−1 􀝔] = ~~  ~~ [sec−1 􀝔] = ~~   ~~ [cos−1 􀝑] = ~~  ~~ [sec−1 􀝑] = ~~   ~~  \n􀝀􀝔 √1 − 􀝔 2 􀝀􀝔 |􀝔|√􀝔2 − 1 􀝀􀝔 √1 − 􀝑 2 􀝀􀝔 |􀝑|√􀝑2 − 1  \n􀝀􀝀􀝔 [tan−1 􀝔] = 1~~ ~~􀝔~~ ~~2~~ ~~ ~~ ~~􀝀􀝀􀝔 [cot−1 􀝔] = 1~~ ~~1􀝔2~~ ~~ ~~ ~~􀝀􀝀􀝔 [tan−1 􀝑] = 1~~ ~~′􀝑~~ ~~2~~ ~~ ~~ ~~􀝀􀝀􀝔 [cot−1 􀝑] = 1~~ ~~􀝑􀝑′~~ ~~2","cbCainZmUOWaLUdr","https://ap.wps.com/l/cbCainZmUOWaLUdr","pdf",207422,"English","# Derivative Rules\n## Constant, Sum, Difference\n## Product and Quotient\n## Power and Chain Rules\n# Common Function Families\n## Exponential and Logarithmic\n## Absolute Value\n## Trigonometric Functions\n## Inverse Trigonometric Functions","[{\"question\":\"What derivative rules are included besides the power rule?\",\"answer\":\"The document lists constant rule, basic sum rule, difference rule, product rule, and quotient rule, in addition to the power rule and chain rule structures.\"},{\"question\":\"How does the chain rule affect the derivative formulas shown?\",\"answer\":\"It provides corresponding derivative versions that multiply by the derivative of the inside function, compared to formulas without chain rule.\"},{\"question\":\"What function types are covered for special formulas?\",\"answer\":\"It includes exponential functions, natural logarithms and other logarithms, absolute value, trigonometric functions, and inverse trigonometric functions.\"}]","DERIVATIVE FORMULAS - derivative rules and applications | PDF"]