[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-669659-105":59,"doc-detail-669659-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","limits_and_derivatives_formulas-limits-and-derivatives-formulas-formulas","limits_and_derivatives_formulas - Limits and Derivatives Formulas - formulas","","A concise reference of calculus formulas for limits and derivatives. It summarizes fundamental limit properties, including operations on functions and standard limit evaluations such as those involving exponentials and trigonometric expressions. It also lists common derivative rules (sum, product, quotient, power, and chain) and derivative formulas for key elementary functions. Finally, it provides definitions and Leibniz-style formulas for higher-order derivatives, along with important closed-form results for derivatives of common function families.",{"@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/limits_and_derivatives_formulas-limits-and-derivatives-formulas-formulas/669659/",{"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/limits_and_derivatives_formulas-limits-and-derivatives-formulas-formulas/669659.png","ImageObject",300,407,{"name":92,"@type":93},"Rhys","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 are the basic properties of limits used to simplify expressions?","Question",{"text":112,"@type":113},"The document states that if lim f(x)=l and lim g(x)=m, then lim(f(x)±g(x))=l±m and lim(f(x)·g(x))=l·m, with additional constant-multiple and reciprocal-style relationships under specified conditions.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"Which derivative rules are included for common differentiation?",{"text":117,"@type":113},"It includes linearity, the product rule, quotient rule, power rule, and chain rule, along with a list of derivatives for elementary functions such as sin, cos, tan, sec, csc, and cot.",{"name":119,"@type":110,"acceptedAnswer":120},"How does the document express higher-order derivatives and product differentiation?",{"text":121,"@type":113},"It defines the second derivative and higher-order derivatives, and provides Leibniz formulas for derivatives of products, including general combinatorial terms for (f·g)^(n).","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},669659,1793563444,{"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":14,"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":24},687207024643,"https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2","[www.mathportal.org](www.mathportal.org)  \nLimits and Derivatives Formulas  \n1. Limits  \nProperties  \nif lx f (x) = l and lx g (x) = m , then lx[ f (x) ± g(x) ] = l ± m  \nlx [ f (x) ⋅ g(x) ] = l ⋅ mlim  f (x) =  l  where m ≠ 0 x→a g (x) m  \na c ⋅ f (x) = c ⋅l  \n1 1 lim ~~ ~~ = ~~ ~~ where l ≠ 0  \nx→a f (x) l  \nFormulas  \n􀀁  1  􀀂 nlx 􀀅􀀃 1+ n 􀀆􀀄 = e  \n1 lim (1+ n ) n = ex→∞  \nlim  sin x = 1  \nx→0 x  \nlim  tan x = 1  \nx→0 x  \ncos x −1  \nlim ~~ ~~ = 0  \nx→0 x  \nlim  xn − an  = nan−1 x→a x − a  \nan −1 lim ~~ ~~ = lnax→0 x  \n2. Common Derivatives  \nBasic Properties and Formulas  \n′  \n(cf ) = cf ′(x)  \n′  \n( f ± g ) = f ′(x) + g ′(x) Product rule  \n′  \n( f ⋅ g ) = f ′ ⋅ g + f ⋅ g ′ Quotient rule  \n′  \n􀀁  f 􀀂  f ′ ⋅ g − f ⋅ g ′  \n􀀅􀀃 g 􀀆􀀄 = g 2  \nPower ruleddx ( xn ) = nxn−1 Chain rule  \nddx ( f (g (x ))) = f ′( g(x) ) g ′(x)  \nCommon Derivatives  \nddx (c ) = 0  \nddx (x ) = 1 ddx~~ ~~ (sin x ) = cos xddx (cos x ) = − sin x  \nddx (tan x ) = co1s2~~ ~~x = sec2 xddx (sec x ) = sec x tan xddx (cscx ) = − csc cot xddx (cot x ) = − ~~ ~~1sin2~~ ~~x = − csc2 xddx ( sin −1 x ) = 1~~ ~~x~~  ~~ d  ( cos−1 x ) = −  1   \ndx 1 − x2  \nddx ( tan−1 x ) = 1~~ ~~+1x2 ddx ( ax ) = ax ln a  \nddx~~ ~~ ( ex ) = exddx (ln x ) = 1x~~ ~~ , x > 0  \nddx (ln x ) = ~~ ~~1x , x ≠ 0  d  (log x ) =  1  , x > 0 dx a xlna  \n[www.mathportal.org](www.mathportal.org)  \n3. Higher-order Derivatives  \nDefinitions and properties  \nSecond derivative  \nd 􀀁 dy 􀀂 d 2 y f ′′ = dx  􀀅􀀃 dx  􀀆􀀄 −  dx2  Higher-Order derivative  \n′ f (n ) = ( f (n−1) )  \n( f + g )(n ) = f (n ) + g (n )  \n( f − g )(n ) = f (n ) − g (n ) Leibniz’s Formulas  \n′′  \n( f ⋅ g ) = f ′′ ⋅ g + 2 f ′ ⋅ g ′ + f .g ′′  \n′′′  \n( f ⋅ g ) = f ′′′ ⋅ g + 3 f ′′ ⋅ g ′ + 3 f ′ ⋅ g ′′ + f ⋅ g ′′′  \n( f ⋅ g )(n ) = f (n ) g + nf (n−1) g + ~~ ~~n~~ ~~(n1~~ ~~−2~~ ~~1) f (n−2 ) g ′′ + ... + fg(n )  \nImportant Formulas  \n( xm )(n ) =  m!  xm−n  \n(m − n ) !( xn )(n ) = n!  \n(loga x )(n ) = (−1x)nn−~~ ~~1~~ ~~l(nn~~ ~~1)~~ ~~!(ln x )(n ) =  (−1)n−1 (n −1) !  \nxn ( ax )(n ) = ax ln n a  \n( ex )(n ) = ex  \n( amx )(n ) = mnamx lnn a (sin x )(n ) = sin 􀀁􀀃 x +  nπ  􀀂􀀄  \n􀀅 2 􀀆  \n(cos x )(n ) = cos 􀀁􀀃 x +  nπ  􀀂􀀄  \n􀀅 2 􀀆","cbCairaz3uE4YYT8","https://ap.wps.com/l/cbCairaz3uE4YYT8","pdf",53462,"English","# 1. Limits\n## Properties\n## Formulas\n# 2. Common Derivatives\n## Basic Properties and Formulas\n## Common Derivatives\n# 3. Higher-order Derivatives\n## Definitions and properties\n## Important Formulas","[{\"question\":\"What are the basic properties of limits used to simplify expressions?\",\"answer\":\"The document states that if lim f(x)=l and lim g(x)=m, then lim(f(x)±g(x))=l±m and lim(f(x)·g(x))=l·m, with additional constant-multiple and reciprocal-style relationships under specified conditions.\"},{\"question\":\"Which derivative rules are included for common differentiation?\",\"answer\":\"It includes linearity, the product rule, quotient rule, power rule, and chain rule, along with a list of derivatives for elementary functions such as sin, cos, tan, sec, csc, and cot.\"},{\"question\":\"How does the document express higher-order derivatives and product differentiation?\",\"answer\":\"It defines the second derivative and higher-order derivatives, and provides Leibniz formulas for derivatives of products, including general combinatorial terms for (f·g)^(n).\"}]","limits_and_derivatives_formulas - Limits and Derivatives Formulas - formulas | PDF"]