[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-571560-105":3,"detail-sidebar-cat-1-en-105":81,"doc-detail-571560-en":127},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":74,"head_meta":76,"extra_data":78,"updated_unix":80},105,"en","common-derivatives-reducedpdf","Common Derivatives - Reduced.pdf","","A condensed calculus reference compiling key differentiation and integration rules alongside common derivative and integral results. It covers basic properties such as constant multiples, sums/differences, product, quotient, and chain rule, then lists polynomial, exponential/logarithmic, trigonometric, and hyperbolic formulas. It similarly provides core integral properties and standard antiderivatives, followed by inverse trig and hyperbolic integral rules and miscellaneous identities. The guide also summarizes standard integration techniques: substitution, integration by parts, trig substitutions, and partial fractions.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/template/","Template",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/template/general/","General",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/template/common-derivatives-reducedpdf/571560/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/common-derivatives-reducedpdf/571560.png","ImageObject",442,249,{"name":42,"@type":43},"Maya Linwood","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-25","2026-10-13",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",10,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"What are the main differentiation rules included in the guide?","Question",{"text":63,"@type":64},"The guide lists constant multiple, sum/difference, power rule, product rule, quotient rule, and chain rule for derivatives.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"Which categories of common derivatives and integrals are provided?",{"text":68,"@type":64},"Common results are grouped into polynomials, exponential/logarithmic functions, trigonometric functions, hyperbolic functions, inverse trigonometric functions, and miscellaneous forms.",{"name":70,"@type":61,"acceptedAnswer":71},"What standard integration techniques does the guide summarize?",{"text":72,"@type":64},"It summarizes substitution, integration by parts, trig substitutions (for specific root forms), and partial fractions (using partial fraction decomposition rules).","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},571560,1792000179,{"code":4,"msg":82,"data":83},"success",[84,89,94,99,104,109,114,119,124],{"id":85,"doc_module":22,"doc_module_name":25,"category_name":86,"show_sort_weight":87,"slug":88},11,"Presentations",90,"presentations",{"id":90,"doc_module":22,"doc_module_name":25,"category_name":91,"show_sort_weight":92,"slug":93},12,"Resumes",80,"resumes",{"id":95,"doc_module":22,"doc_module_name":25,"category_name":96,"show_sort_weight":97,"slug":98},14,"Invoices",70,"invoices",{"id":100,"doc_module":22,"doc_module_name":25,"category_name":101,"show_sort_weight":102,"slug":103},15,"Posters",60,"posters",{"id":105,"doc_module":22,"doc_module_name":25,"category_name":106,"show_sort_weight":107,"slug":108},16,"Social Media",50,"social-media",{"id":110,"doc_module":22,"doc_module_name":25,"category_name":111,"show_sort_weight":112,"slug":113},17,"Forms",40,"forms",{"id":115,"doc_module":22,"doc_module_name":25,"category_name":116,"show_sort_weight":117,"slug":118},18,"Letters",30,"letters",{"id":120,"doc_module":22,"doc_module_name":25,"category_name":121,"show_sort_weight":122,"slug":123},21,"Paper Templates",5,"papers-templates",{"id":125,"doc_module":22,"doc_module_name":25,"category_name":29,"show_sort_weight":4,"slug":126},158,"general-158",{"code":4,"msg":82,"data":128},{"doc_id":79,"user_id":129,"nickname":42,"user_avatar":130,"doc_module":22,"category_id":125,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":131,"file_id":132,"file_url":133,"file_type":134,"file_size":135,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":26,"language":136,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":137,"faqs":138,"seo_title":139,"seo_description":12,"update_tm":140,"read_time":22},962084928432,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","Derivatives  \nBasic Properties/Formulas/Rules  \nddx ( cf ( x ) ) = cf ¢ (x ) , c is any constant. ( f ( x ) ± g (x ))¢ = f ¢ (x ) ± g¢ (x ) ddx ( xn) = nxn-1 , n is any number. ~~ ~~ddx ( c ) = 0 , c is any constant.  \n( f g )¢ = f ¢ g + f g ¢ –(Product Rule) ÁÊ  f ˆ˜¢ = ~~ ~~f~~ ~~¢~~ ~~g-2f~~ ~~g~~ ~~¢ –(Quotient Rule)  \nË g ¯ g  \nddx ( f ( g ( x ))) = f ¢ (g ( x )) g ¢ (x ) (Chain Rule)  \nddx (eg(x ) ) = g ¢ (x ) eg (x ) ~~ ~~ddx (ln g ( x )) = ~~ ~~gg¢~~(~~(xx~~)~~)  \nCommon Derivatives  \nPolynomials  \nddx~~ ~~ ( c ) = 0 ~~ ~~ddx ( x ) = 1 ~~ ~~ddx ( cx) = c ~~ ~~ddx~~ ~~ ( xn) = nxn-1 ~~ ~~ddx~~ ~~ ( cxn) = ncxn-1  \nExponential/Logarithm Functions  \nddx~~ ~~ ( ax) = ax ln ( a ) ddx~~ ~~ (ln ( x )) = 1x , x > 0  \nHyperbolic Trig Functions ddx~~ ~~ ( sinh x) = cosh xddx~~ ~~ ( sech x) = -sech x tanh x  \nddx ( ex) = exddx (ln x ) = 1x~~ ~~ , x 􀂹 0  \nddx ( cosh x) = sinh xddx ( csch x) = -csch x coth x  \nddx ( tan x) = sec2 xddx ( cot x) = -csc2 x  \nddx ( tan-1 x ) = 1~~ ~~+1x2 ddx ( cot-1 x ) = - ~~ ~~1~~ ~~+1x2  \nddx (loga ( x )) = ~~ ~~x~~l~~1n~~ ~~a~~ ~~ , x > 0  \nddx ( tanh x) = sech2 xddx ( coth x) = -csch2 x  \nVisit [http://tutorial. math.lamar.edu](http://tutorial. math.lamar.edu for a complete)[ for a complete](http://tutorial. math.lamar.edu for a complete) set of Calculus I & II notes. © 2005 Paul Dawkins  \nIntegrals  \nBasic Properties/Formulas/Rules  \nÚcf ( x ) dx = c Ú f ( x ) dx , c is a constant.  \nÚbaf ( x ) dx = F ( x )ba = F (b )-F ( a ) where  \nÚba cf (x ) dx = c Úba f ( x ) dx , c is a constant.  \nÚaa f ( x ) dx = 0  \nÚba f ( x ) dx = Úca f ( x ) dx + Úcbf ( x ) dx  \nÚ f ( x ) ± g ( x ) dx = Ú f ( x ) dx ± Ú g ( x ) dx F ( x ) = Ú f ( x ) dx  \nÚbaf ( x ) ± g (x ) dx = Úbaf (x ) dx ± Úba g ( x ) dx  \nÚbaf ( x ) dx = - Úab f ( x ) dx  \nÚbac dx = c ( b-a )  \nIf f ( x ) 􀂳 0 on a £ x £ b then Úba f ( x ) dx 􀂳 0  \nIf f ( x ) 􀂳 g ( x ) on a £ x £ b then Úbaf ( x ) dx 􀂳 Úbag (x ) dx  \nCommon Integrals  \nPolynomials  \nÚ dx = x + c  \nıÛÙ ~~ ~~1x dx = ln x + c  \nÚ k dx = k x + c  \nÚx-1 dx = ln x + c  \nÚxndx = ~~ ~~n~~ ~~~~ ~~1 xn+1 + c, n 􀂹 -1  \nÚx-ndx = -n1+1x-n+1 + c , n 􀂹 1  \nıÛÙ ~~ ~~ax~~ ~~b dx = 1a ln  \nax + b + c  \nÚxpqdx = ~~  ~~pq~~ ~~~~ ~~1 x~~ ~~pq+1 + c = ~~ ~~p~~ ~~~~ ~~q xp+qq + c  \nTrig Functions  \nÚcosu du = sin u + c  \nÚsecu tanu du = secu + c  \nÚsin u du = -cosu + c  \nÚcscu cotudu = -cscu + c  \nÚsec2 u du = tan u + c  \nÚcsc2 u du = -cot u + c  \nÚ tan u du = ln sec u + c  \nÚsec u du = lnsec u + tan u + c  \nÚcscu du = lncscu-cot u + c  \nÚcot u du = ln sin u + c  \nÚsec3 u du = ~~ ~~12~~ ~~ ( secu tan u + ln sec u + tan u) + c  \nÚcsc3 u du = ~~ ~~12 (-cscu cot u + ln cscu-cot u) + c  \nExponential/Logarithm Functions  \nÚ eu du = eu + c Úau du = ~~ l~~anua~~ ~~ + c  \nÚeau sin( bu) du = ~~ ~~a2eb2~~ ~~ ( a sin( bu)-b cos( bu)) + c  \nÚeau cos ( bu) du = a2eb2~~ ~~ ( acos( bu) + b sin ( bu))+ c  \nÚ ln u du = u ln ( u ) -u + c  \nÚueu du = ( u-1) eu + c  \nıÛÙ ~~ ~~u~~ ~~l1n~~ ~~u~~ ~~ du = ln ln u + c  \nVisit  [http://tutorial. math.lamar.edu](http://tutorial. math.lamar.edu for a complete)[ for a complete](http://tutorial. math.lamar.edu for a complete) set of Calculus I & II notes. © 2005 Paul Dawkins  \nInverse Trig Functions  \nıÙÛ ~~ ~~1-~~  ~~ du = sin-1 ÁÊ ~~ ~~uËa~~ ~~ ˆ¯˜ + c  \nÛ 1 1 1 Ê u ˆ  \nıÙ  a2 + u2  du = ~~ ~~a~~ ~~ tan- ËÁ ~~ ~~a~~ ~~ ¯˜ + c  \nÛ 1 1 -1 Ê u ˆ  \nÙ ~~    ~~2~~ ~~2~~  ~~ du = ~~ ~~ sec Á  ˜ + c ı u a Ëa ¯  \nÚsin-1u du = usin-1 u +  + c  \nÚ tan-1u du = u tan-1 u- ~~ ~~12 ln (1+ u2 ) + c  \nÚcos-1u du = u cos-1 u-  + c  \nHyperbolic Trig Functions  \nÚsinh u du = cosh u + c  \nÚsech tanhu du = -sech u + c  \nÚ tanh u du = ln ( cosh u) + c  \nÚcosh u du = sinh u + c  \nÚcsch coth u du = -csch u + c  \nÚsech u du = tan-1 sinh u + c  \nÚsech 2 u du = tanh u + c  \nÚcsch2 u du = -coth u + c  \nMiscellaneous  \nıÛÙ ~~ ~~a2-1~~ ~~u2~~ ~~ du = ~~ ~~21a ln  ~~ ~~uu~~ ~~+-aa + c ıÛÙ ~~ ~~u2-1~~ ~~a2~~ ~~ du = 21a~~ ~~ ln  \n2  \na2 + u2 du = u~~ ~~ a2 + u2  + ~~ ~~a~~ ~~ln u + a2 + u2   + c  \nÚ  2 22  \nÚ du","cbCaipHV06Xk8ohc","https://ap.wps.com/l/cbCaipHV06Xk8ohc","pdf",139629,"English","# Derivatives\n## Basic Properties, Formulas, and Rules\n## Common Derivatives\n# Integrals\n## Basic Properties, Formulas, and Rules\n## Common Integrals\n# Standard Integration Techniques\n## Substitution\n## Integration by Parts\n## Trig Substitutions\n## Partial Fractions","[{\"question\":\"What are the main differentiation rules included in the guide?\",\"answer\":\"The guide lists constant multiple, sum/difference, power rule, product rule, quotient rule, and chain rule for derivatives.\"},{\"question\":\"Which categories of common derivatives and integrals are provided?\",\"answer\":\"Common results are grouped into polynomials, exponential/logarithmic functions, trigonometric functions, hyperbolic functions, inverse trigonometric functions, and miscellaneous forms.\"},{\"question\":\"What standard integration techniques does the guide summarize?\",\"answer\":\"It summarizes substitution, integration by parts, trig substitutions (for specific root forms), and partial fractions (using partial fraction decomposition rules).\"}]","Common Derivatives - Reduced.pdf | PDF",1791922362]