[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-571562-105":53,"doc-detail-571562-en":127},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":120,"head_meta":122,"extra_data":124,"updated_unix":126},105,"en","table-of-derivatives-indefinite-integrals-properties-of-exponentials-properties-of-logarithms","Table of Derivatives - Indefinite Integrals - Properties of Exponentials - Properties of Logarithms","","A compact reference covering derivative and indefinite integral rules with constant parameters. It lists standard differentiation results for linear combinations, products, quotients, powers, and common functions such as trigonometric and inverse trigonometric forms, logarithms, and exponentials. It also provides key exponential identities (powers, sums, negatives, derivatives, integrals, limits, and graph notes) and logarithm properties for positive real inputs, including base-change style relations, product/quotient rules, differentiation and integral formulas, and limit behavior.",{"@graph":63,"@context":119},[64,80,102],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":51,"@type":70,"position":76},"https://docshare.wps.com/template/general/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/table-of-derivatives-indefinite-integrals-properties-of-exponentials-properties-of-logarithms/571562/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/table-of-derivatives-indefinite-integrals-properties-of-exponentials-properties-of-logarithms/571562.png","ImageObject",442,249,{"name":88,"@type":89},"Aria Callaghan","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-23","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":101},"InteractionCounter",{"@type":100},"ViewAction",7,{"@type":103,"mainEntity":104},"FAQPage",[105,111,115],{"name":106,"@type":107,"acceptedAnswer":108},"What assumptions are used for variables and constants in the derivative and integral tables?","Question",{"text":109,"@type":110},"In the tables, a and b are constants independent of x. The integral table also includes an arbitrary constant C.","Answer",{"name":112,"@type":107,"acceptedAnswer":113},"Which differentiation rules are included for algebraic combinations of functions?",{"text":114,"@type":110},"The table covers differentiation of linear combinations af(x)+bg(x), products f(x)g(x), and expressions involving sums of powers such as f(x)/g(x) patterns, along with related derived forms.",{"name":116,"@type":107,"acceptedAnswer":117},"What key exponential identities and derivative/integral rules are provided?",{"text":118,"@type":110},"The exponential section lists identities like e^{x+y}=e^x e^y and a^{x+y}=a^x a^y, negative exponent relations, derivative rules such as d/dx e^x=e^x and d/dx a^x=(ln a)a^x, and corresponding integral rules.","https://schema.org",{"og:url":78,"og:type":121,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":123,"canonical":78},"index,follow",{"doc_id":125,"site_id":56},571562,1792009244,{"code":4,"msg":5,"data":128},{"doc_id":125,"user_id":129,"nickname":88,"user_avatar":130,"doc_module":9,"category_id":50,"category_name":51,"doc_title":59,"doc_description":61,"doc_content":131,"file_id":132,"file_url":133,"file_type":134,"file_size":135,"view_count":101,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":79,"language":136,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":137,"faqs":138,"seo_title":139,"seo_description":61,"update_tm":140,"read_time":73},962084926284,"https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0","Table of Derivatives  \nThroughout this table, a and b are constants, independent of x.  \n\n| F (x) | F 0 (x) = ~~d~~dFx |\n| --- | --- |\n| af (x) + bg(x) f (x) + g(x)\u003Cbr>f (x) 􀀀 g(x)\u003Cbr>af (x) | af 0 (x) + bg0 (x)\u003Cbr>f 0 (x) + g0 (x)\u003Cbr>f 0 (x) 􀀀 g0 (x)\u003Cbr>af 0 (x) |\n| f (x)g (x) | f 0 (x)g (x) + f(x)g 0 (x) |\n| f (x)g (x)h (x) | f 0 (x)g (x)h (x) + f(x)g 0 (x)h (x) + f(x)g (x)h0 (x) |\n| f (x)\u003Cbr>g (x) | f 0 (x)g (x)􀀀f (x)g 0 (x)\u003Cbr>g (x)2 |\n|  1 \u003Cbr>g (x) | g 0 (x)\u003Cbr>􀀀\u003Cbr>g (x)2 |\n| f 􀀀g (x)􀀁 | f 0 􀀀g (x)􀀁g0 (x) |\n| 1 | 0 |\n| a | 0 |\n| xa | axa􀀀1 |\n| g (x)a | ag (x)a􀀀1g 0 (x) |\n| sin x | cos x |\n| sin g(x) | g 0 (x) cosg(x) |\n| cos x | 􀀀 sin x |\n| cosg(x) | 􀀀g0 (x) sin g(x) |\n| tan x | sec2 x |\n| csc x | 􀀀 csc x cotx |\n| sec x | sec x tan x |\n| cotx | 􀀀 csc2 x |\n| ex | ex |\n| eg (x) | g 0 (x)eg (x) |\n| ax | (ln a) ax |\n| ln x\u003Cbr>ln g(x)\u003Cbr>loga x | 1 xg 0 (x)\u003Cbr>g (x)  1  x ln a |\n| arcsin x\u003Cbr>arcsing(x)\u003Cbr>arccos x\u003Cbr>arctan x\u003Cbr>arctang(x)\u003Cbr>arccscx\u003Cbr>arcsec x\u003Cbr>arccotx | 1\u003Cbr>p 1􀀀x2  g 0 (x)  p1􀀀g(x)2 􀀀  1 \u003Cbr>p 1􀀀x2  1  1+x2  g 0 (x)  1+g(x)2 􀀀  1 \u003Cbr>xp 1􀀀x2  1 \u003Cbr>xp 1􀀀x2 􀀀  1  1+x2 |\n\nTable of Inde􀀌nite Integrals  \nThroughout this table, a and b are given constants, independent of x and C is an arbitrary constant.  \n\n| f (x) | F (x) = R f (x) dx |\n| --- | --- |\n| af (x) + bg(x) f (x) + g(x)\u003Cbr>f (x) 􀀀 g(x)\u003Cbr>af (x) | a R f (x) dx + b R g(x) dx + CR f (x) dx +R g(x) dx + C\u003Cbr>R f (x) dx 􀀀R g(x) dx + CaR f (x) dx + C |\n| u (x)v 0 (x) | u (x)v (x) 􀀀R u0 (x)v (x) dx + C |\n| f 􀀀y (x)􀀁y0 (x) | F 􀀀y(x)􀀁 where F(y) = R f (y) dy |\n| 1 | x + C |\n| a | ax + C |\n| xa | ~~x~~a11 + C if a  􀀀1 |\n| 1 x | ln j xj + C |\n| g (x)ag 0 (x) | g~~ ~~(xa+)a + C if a  􀀀1 |\n| sin x | 􀀀 cos x + C |\n| g 0 (x) sin g(x) | 􀀀 cosg(x) + C |\n| cos x | sin x + C |\n| tan x | ln j sec xj + C |\n| cscx | ln j csc x 􀀀 cot xj + C |\n| sec x | ln j sec x + tan xj + C |\n| cotx | ln j sin xj + C |\n| sec2 x | tan x + C |\n| csc2 x | 􀀀 cotx + C |\n| sec x tan x | sec x + C |\n| csc x cotx | 􀀀 cscx + C |\n| ex | ex + C |\n| eg (x)g 0 (x) | eg (x) + C |\n| eax | ~~1~~a eax + C |\n| ax | l~~1~~na ax + C |\n| ln x | xlnx 􀀀 x + C |\n| 1\u003Cbr>p 1􀀀x2  g 0 (x)  p1􀀀g(x)2 1\u003Cbr>pa2 􀀀x2  1  1+x2\u003Cbr> g 0 (x)  1+g(x)2\u003Cbr> 1  a2 +x2\u003Cbr> 1 \u003Cbr>xp 1􀀀x2 | arcsin x + C\u003Cbr>arcsing(x) + C\u003Cbr>arcsin ~~x~~a + C\u003Cbr>arctan x + C\u003Cbr>arctang(x) + C\u003Cbr>~~1~~a arctan ~~x~~a + C\u003Cbr>arcsec x + C |\n\nProperties of Exponentials  \nIn the following, x and y are arbitrary real numbers, a and b are arbitrary constants that are strictly bigger than zero and e is 2.7182818284, to ten decimal places.  \n1) e0 = 1, a0 = 1  \n2) ex+y = ex ey , ax+y = ax ay  \n3) e 􀀀x = e , a 􀀀x = a  \n4) 􀀀ex 􀀁 y = exy , 􀀀ax 􀀁 y = axy  \n5) ~~d~~dx ex = ex , ~~ d~~dx eg (x) = g 0 (x)eg (x) , ~~ d~~dx ax = (ln a) ax  \n6) R ex dx = ex + C , R eax dx = ~~1~~aeax + C if a  0  \n7) lim ex = 1 , lim ex = 0  \nx!1 x!􀀀1  \nlim ax = 1 , lim ax = 0 if a > 1  \nx!1 x!􀀀1  \nlim ax = 0, lim ax = 1 if 0 \u003C a \u003C 1  \nx!1 x!􀀀1  \n8) The graph of 2x is given below. The graph of ax , for any a > 1, is similar.  \ny  \n6 ~~ ~~  \n4 ~~ ~~  \n2 ~~ ~~  \n1   \n􀀀3 􀀀2 􀀀1  \ny = 2x  \nx  \n1 2 3  \nProperties of Logarithms  \nIn the following, x and y are arbitrary real numbers that are strictly bigger than 0, a isan arbitrary constant that is strictly bigger than one and e is 2.7182818284, to ten decimal places.  \n1) elnx = x, aloga x = x, loge x = ln x, loga x = ~~l~~l~~n~~n~~x~~a  \n2) loga 􀀀ax 􀀁 = x, ln 􀀀ex 􀀁 = x ln 1 = 0, loga 1 = 0  \nln e = 1, loga a = 1  \n3) ln(xy) = ln x + ln y , loga (xy) = loga x + loga y  \n4) ln 􀀀 ~~x~~y 􀀁 = ln x 􀀀 ln y , loga 􀀀 ~~x~~y 􀀁 = loga x 􀀀 loga y  \nln 􀀀 ~~1~~y 􀀁 = 􀀀 ln y , loga 􀀀 ~~1~~y 􀀁 = 􀀀 loga y ,  \n5) ln(xy ) = y lnx, loga (xy ) = y loga x  \n6) ~~d~~dx lnx = ~~1~~x , ~~d~~dx ln(g(x)) = gg0((xx)) , ~~d~~dx loga x = xl~~1~~n~~ ~~a  \n7) R ~~1~~xdx = ln jxj + C , R lnx dx = xlnx 􀀀 x + C  \n8) lim ln x = 1 , lim ln x = 􀀀1 x!1 x!0  \nxl loga x = 1 , lxim!0 loga x = 􀀀1  \n9) The graph of ln x is given below. The graph of loga x, for any a > 1, is simi","cbCaieNf6tmX3LVz","https://ap.wps.com/l/cbCaieNf6tmX3LVz","pdf",57498,"English","# Table of Derivatives\n# Table of Inde􀀌nite Integrals\n# Properties of Exponentials\n# Properties of Logarithms","[{\"question\":\"What assumptions are used for variables and constants in the derivative and integral tables?\",\"answer\":\"In the tables, a and b are constants independent of x. The integral table also includes an arbitrary constant C.\"},{\"question\":\"Which differentiation rules are included for algebraic combinations of functions?\",\"answer\":\"The table covers differentiation of linear combinations af(x)+bg(x), products f(x)g(x), and expressions involving sums of powers such as f(x)/g(x) patterns, along with related derived forms.\"},{\"question\":\"What key exponential identities and derivative/integral rules are provided?\",\"answer\":\"The exponential section lists identities like e^{x+y}=e^x e^y and a^{x+y}=a^x a^y, negative exponent relations, derivative rules such as d/dx e^x=e^x and d/dx a^x=(ln a)a^x, and corresponding integral rules.\"}]","Table of Derivatives - Indefinite Integrals - Properties of Exponentials - Properties of Logarithms | PDF",1791922363]