[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572422-105":53,"doc-detail-572422-en":126},{"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":119,"head_meta":121,"extra_data":123,"updated_unix":125},105,"en","basic-derivatives-and-integrals-tables-key-rules","Basic Derivatives and Integrals Tables - Key Rules","","Reference tables compile core differentiation and integration rules for functions u(x) and parameters r, a, b, and constant ɸ (with stated constraints). Derivative sections present the power rule, trigonometric rules, hyperbolic rules, exponential/logarithmic rules, inverse trigonometric functions, inverse hyperbolic functions, and product/quotient rules, each paired with worked examples. Integral sections list corresponding power rule results and multiple families for trigonometric, additional trigonometric, and hyperbolic integrals, organized for quick substitution and problem solving.",{"@graph":63,"@context":118},[64,80,101],{"@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/basic-derivatives-and-integrals-tables-key-rules/572422/",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/basic-derivatives-and-integrals-tables-key-rules/572422.png","ImageObject",442,249,{"name":88,"@type":89},"lilac","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-19","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":79},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"What does the Power Rule state for derivatives?","Question",{"text":108,"@type":109},"It gives the differentiation result for d{(u)^n}/dx type expressions in terms of n and u'(x), assuming u is differentiable.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"Which set of trigonometric rules are covered for derivatives?",{"text":113,"@type":109},"It includes rules for sin(u), cos(u), tan(u), cot(u), sec(u), and csc(u), each expressed using u'(x) and related trigonometric functions.",{"name":115,"@type":106,"acceptedAnswer":116},"What product and quotient differentiation identities are provided?",{"text":117,"@type":109},"It lists the derivative of uv as u'v + uv' and the quotient-type derivative form using u and v with their derivatives, each shown with examples.","https://schema.org",{"og:url":78,"og:type":120,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":122,"canonical":78},"index,follow",{"doc_id":124,"site_id":56},572422,1792011104,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":50,"category_name":51,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":79,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":135,"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},3985747861865,"https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0","A: TABLE OF BASIC DERIVATIVES Let u = u(x) be a differentiable function of the independent variable x , that is u′(x) exists.  \n\n| (A) The Power Rule : | Examples : |\n| --- | --- |\n|  d  {un } = nun−1 u ′\u003Cbr>dx . |  d  {(x3 + 4x + 1)3/4} = 3  (x3 + 4x + 1)−1/4 (3x2 + 4)\u003Cbr>dx 4 . |\n|  d  {  u } =  1  u ′\u003Cbr>dx 2 u .\u003Cbr>|  d  { 2 4x + 7x } =  1  (−8x + 35x4 )\u003Cbr>dx 2 2 − 4x2 + 7x5 |\n| ddx~~ ~~ {c} = 0 , c is a constant | ddx~~ ~~ {􀀁6 } = 0 , since 􀀁 ≅ 3. 14 is a constant. |\n\n\n| (B) The Six Trigonometric Rules : | Examples : |\n| --- | --- |\n| ddx~~ ~~ {sin(u)} = cos(u) . u ′ | ddx~~ ~~ {sin(x3 )} = cos(x3 ) . 3x2 |\n| ddx~~ ~~ {cos(u)} = − sin(u) . u ′ |  d  {cos  x )} = − sin(  x )  1 \u003Cbr>dx . 2 x\u003Cbr>|\n| ddx~~ ~~ {tan(u)} = sec 2 (u) . u ′ |  d  {tan{ 5 )} = sec 2 (5x−2) (−10x−3)\u003Cbr>dx x 2 . |\n| ddx~~ ~~ {cot(u)} = − csc2 (u) . u ′ | ddx~~ ~~ [cot{sin(2x)}] = − csc2 {sin(2x)} . 2 cos(2x) . |\n| ddx~~ ~~ {sec(u)} = sec(u)tan(u) . u ′ |  d  {sec( 4 x )} = sec( 4 x ) tan( 4 x )  1  x−3/4\u003Cbr>dx . 4 |\n| ddx~~ ~~ {csc(u)} = − csc(u)cot(u) . u ′ | ddx~~ ~~ {csc(8x − 7)} = − csc(8x − 7)cot(8x − 7). 8 |\n\n\n| (C) The Six Hyperbolic Rules : | Examples : |\n| --- | --- |\n| ddx~~ ~~ {sinh(u)} = cosh(u) . u ′ |  d  {sinh( 3 x )} = cosh( 3 x )  1  x−2/3\u003Cbr>dx . 3\u003Cbr>|\n| ddx~~ ~~ {cosh(u)} = sinh(u) . u ′ | ddx~~ ~~ {cosh(sec(x)} = sinh{sec(x)} . sec(x)tan(x) |\n| ddx~~ ~~ {tanh(u)} =sech2 (u) . u ′ |  d  [tanh{x3 + sin(x2 )}] =sech2 {x3 + sin(x2 )} (3x2 + 2x cos(x2 ))\u003Cbr>dx . |\n| ddx~~ ~~ {coth(u)} = −csch2 (u) . u ′ | ddx~~ ~~ {coth( ~~ 1x ~~ + 2x)} = −csch2 ( ~~ 1x ~~ + 2x) . (− ~~ ~~x~~1~~2~~ ~~ + 2) |\n| ddx~~ ~~ {sech(u)} = −sech(u)tanh(u) . u ′ | ddx~~ ~~ {sech{9x)} = −sech(9x)tanh(9x). 9 |\n| ddx~~ ~~ {csch(u)} = −csch(u)coth(u) . u ′ | ddx~~ ~~ [csch{sinh(3x)}] = −csch{sinh(3x)} coth{sinh(3x)} . 3 cosh(3x) |\n\n\n| (D) The Exponential & Logarithmic Rule : | Examples : |\n| --- | --- |\n|  d  {eu } = eu u ′\u003Cbr>dx . |  d  {e−x3 } = e−x3 (−3x2 )\u003Cbr>dx . |\n| ddx~~ ~~ {ln| u |} = u~~u~~′~~ ~~ | ddx~~ ~~ {ln| x 3 + 5x + 6 |} = x~~ ~~3~~3~~25~~+~~x~~ ~~~~ ~~6 |\n|  d  {au } = au . ln(a) . u ′ a ∈ R a > 0 , a ≠ 1\u003Cbr>dx , , | ~~ ~~ddx~~ ~~ {2sec(x)} = 2 sec(x). ln(2) . sec(x)tan(x) |\n| ddx~~ ~~ {loga | u |} = ln~~1~~(a)~~ ~~ ~~ ~~u~~u~~′ , a ∈ R , a > 0 , a ≠ 1 |  d   1 sec 2 (x)\u003Cbr>dx {log4 | tan(x) |} = ln(4) tan(x) |\n\n\n| (E) The Six Inverse Trigonometric Functions : | Examples : |\n| --- | --- |\n|  d  {sin−1(u)} =  u ′ \u003Cbr>dx 1 − u 2 |  d  {sin−1(4x2 )} =  8x \u003Cbr>dx 1 − 16x4 |\n|  d  {cos−1(u)} = −  u ′ \u003Cbr>dx 1 − u 2 |  d  {cos−1(3x)} = −  3 \u003Cbr>dx 1 − 9x2 |\n| ddx~~ ~~ {tan−1(u)} = ~~ ~~1~~ ~~′u~~ ~~2~~ ~~ |  d  {tan−1(  x )} =  ~~ ~~2~~ ~~~~ 1~~x~~ ~~  =  1 \u003Cbr>dx 1 + x 2 x (1 + x)\u003Cbr>|\n| ddx~~ ~~ {cot−1(u)} = −~~ ~~1~~ ~~′u~~ ~~2~~ ~~ | ddx~~ ~~ {cot−1(ex )} = −~~ ~~1~~ ~~xe~~ ~~2x~~ ~~ |\n|  d  {sec−1(u)} =  u ′ \u003Cbr>dx |u|u 2 − 1 |  d  [sec−1(x4 )} =  4x3  =  4x3 \u003Cbr>dx |x4 | x 8 − 1 x4 x 8 − 1 |\n|  d  {csc−1(u)} = −  u ′ \u003Cbr>dx |u|u 2 − 1 |  d  {csc−1(2x)} = −  2  = −  1 \u003Cbr>dx |2x|4x2 − 1 |x|4x2 − 1 |\n\n\n| (F) The Inverse Hyperbolic Functions : | Examples : |  |\n| --- | --- | --- |\n|  d  {sinh−1(u)} =  u ′ \u003Cbr>dx 1 + u 2 |  d  {sinh−1(ln(x)} =  1/x \u003Cbr>dx 1 + ln2 (x) |  |\n|  d  {cosh−1(u)} =  u ′ \u003Cbr>dx u 2 − 1 | ddx~~ ~~ {cosh−1(5x)} = |  5 \u003Cbr>25x2 − 1 |\n| ddx~~ ~~ {tanh−1(u)} = ~~ ~~1~~ ~~′u~~ ~~2~~ ~~ | ddx~~ ~~ {tanh−1( ~~ 2x ~~ )} = ~~ ~~12x24x2~~  ~~ = x~~ ~~2−~~ ~~4 |  |\n\n\n| (G) The Product and Quotient Rules : | Examples : |\n| --- | --- |\n| ddx~~ ~~ {uv} = u ′ v + uv ′ | ddx~~ ~~ {x3 ln(5x + 1)} = 3x2 ln(5x + 1) + x 3 ~~ ~~5x~~ ~~1 |\n|  d  {ku} = ku ′ k is a constant\u003Cbr>dx , | ddx~~ ~~ { ~~ ~~x34~~ ~~ } = ~~1  ~~d4~~  ~~dx~~ ~~ {x3 } = ~~1~~4~~ ~~ . 3x2 = ~~3~~x42 |\n| ddx~~ ~~ { ~~ ~~u~~v ~~ } = u~~ ~~′~~ ~~v~~ ~~−v2~~ ~~uv~~ ~~′~~ ~~ | d tan(2x) 2 sec 2 (2x) .x 3 − tan(2x) .3x2\u003Cbr>~~ ~~ {  } = ~~ ~~\u003Cbr>dx x 3 x 6 |\n\nIn table above it is assumed that u = u(x) and v = v(x) are differentiable functions  \nB: TABLE OF BASIC INTEGRALS Let r , a , ","cbCaisgiWoEMv04Z","https://ap.wps.com/l/cbCaisgiWoEMv04Z","pdf",95242,6,"English","# Table of Basic Derivatives\n## The Power Rule\n## The Six Trigonometric Rules\n## The Six Hyperbolic Rules\n## The Exponential & Logarithmic Rule\n## The Six Inverse Trigonometric Functions\n## The Inverse Hyperbolic Functions\n## The Product and Quotient Rules\n# Table of Basic Integrals\n## The Power Rule\n## The Six Trigonometric Rules\n## Additional Trigonometric Rules\n## The Six Hyperbolic Rules","[{\"question\":\"What does the Power Rule state for derivatives?\",\"answer\":\"It gives the differentiation result for d{(u)^n}/dx type expressions in terms of n and u'(x), assuming u is differentiable.\"},{\"question\":\"Which set of trigonometric rules are covered for derivatives?\",\"answer\":\"It includes rules for sin(u), cos(u), tan(u), cot(u), sec(u), and csc(u), each expressed using u'(x) and related trigonometric functions.\"},{\"question\":\"What product and quotient differentiation identities are provided?\",\"answer\":\"It lists the derivative of uv as u'v + uv' and the quotient-type derivative form using u and v with their derivatives, each shown with examples.\"}]","Basic Derivatives and Integrals Tables - Key Rules | PDF",1791927321]