[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572421-105":53,"doc-detail-572421-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","calculus-3-cheat-sheet-key-derivatives-integrals-and-3d-concepts","Calculus 3 Cheat Sheet - Key Derivatives, Integrals, and 3D Concepts","","A compact calculus 3 reference organizing essential derivative rules for trigonometric, inverse trigonometric, hyperbolic, and logarithmic/exponential forms, followed by a set of core integration formulas and common techniques. It summarizes u-substitution using u=f(x), integration by parts, and foundational trig/hyperbolic identities. The guide then covers 3D analytic geometry and vector/vector-product operations, including plane and line equations, distance from a point to a plane, coordinate system conversions, surface equations, and partial derivatives.",{"@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":11,"@type":70,"position":76},"https://docshare.wps.com/template/presentations/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/calculus-3-cheat-sheet-key-derivatives-integrals-and-3d-concepts/572421/",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/calculus-3-cheat-sheet-key-derivatives-integrals-and-3d-concepts/572421.png","ImageObject",442,249,{"name":88,"@type":89},"EvelynWrites","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-22","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":47},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"What derivative rules are included in the cheat sheet?","Question",{"text":108,"@type":109},"It lists derivatives for inverse trig functions, hyperbolic functions (sinh, cosh, tanh, coth, sech, csch), and key trigonometric derivatives such as sin(x), cos(x), tan(x), sec(x), and csc(x).","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"Which integration techniques and formulas does the document provide?",{"text":113,"@type":109},"It provides a set of standard integral forms (including inverse trig-related integrals and logarithmic/arctangent forms) and outlines u-substitution with u=f(x) and integration by parts using ∫u dv = uv − ∫v du.",{"name":115,"@type":106,"acceptedAnswer":116},"What 3D geometry and vector topics are covered?",{"text":117,"@type":109},"It summarizes 3D distance and midpoint formulas, sphere equation, plane and line equations, dot and cross products (including geometric interpretations), coordinate conversions between rectangular/cylindrical/spherical, common surface equations, and the definition of partial derivatives.","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},572421,1792360249,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":8,"category_name":11,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":47,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":73,"language":135,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":136,"faqs":137,"seo_title":138,"seo_description":61,"update_tm":139,"read_time":9},687212989558,"https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","Derivatives  \nDx ex = ex  \nDx sin(x) = cos(x)  \nDx cos(x) = 􀀀 sin(x)  \nDx tan(x) = sec2 (x)  \nDx cot(x) = 􀀀 csc2 (x)  \nDx sec(x) = sec(x) tan(x) Dx csc(x) = 􀀀 csc(x) cot(x) Dx sin 􀀀 1 = p1~~ ~~x2 ; x 2 [􀀀1 ; 1] Dx cos 􀀀 1 = p􀀀~~1~~x2 ; x 2 [􀀀1 ; 1] Dx tan 􀀀 1 = 1+~~1~~x~~2~~ ; 􀀀2~~􀀙~~ 􀀔 x 􀀔 ~~􀀙~~2 Dx sec 􀀀 1 =  1~~ ~~ j xj > 1  \njxj px2 􀀀 1 ; Dx sinh(x) = cosh(x)  \nDx cosh(x) =  \nDx tanh(x) =  \nDx coth(x) =  \nDx sech (x) =  \nDxcsch (x) =  \nsinh(x) sech2 (x)  \n􀀀csch2 (x)  \n􀀀sech(x) tanh(x)  \n􀀀csch(x) coth(x)  \nDx sinh 􀀀 1 = px~~1~~2+1  \nDx cosh 􀀀 1 = p2~~1~~􀀀~~ ~~1 ; x > 1 Dx tanh 􀀀 1 = 1~~ ~~􀀀~~1~~x~~2~~ 􀀀 1 \u003C x \u003C 1 Dx sech 􀀀 1 = xp1~~1~~􀀀~~ ~~x2 ; 0 \u003C x \u003C 1 Dx ln(x) = ~~1~~x  \nIntegrals  \nR ~~1~~x dx = ln j xj + c R ex dx = ex + c R ax dx = ln~~1~~aax + c R eax dx = ~~1~~a eax + c  \nR p1x2 dx = sin 􀀀1(x) + c R 1+~~1~~x~~2~~ dx = tan 􀀀1(x) + c  \nR xp~~1~~x2~~ ~~􀀀~~ ~~1 dx = sec 􀀀1(x) + c R sinh(x)dx = cosh(x) + c R cosh(x)dx = sinh(x) + c R tanh(x)dx = ln j cosh(x)j + c R tanh(x)sech (x)dx = 􀀀sech(x) + c R sech2 (x)dx = tanh(x) + c R csch (x) coth(x)dx = 􀀀csch(x) + c R tan(x)dx = 􀀀 ln j cos(x)j + c R cot(x)dx = ln j sin(x)j + c R cos(x)dx = sin(x) + c R sin(x)dx = 􀀀 cos(x) + c  \nR pa2~~1~~􀀀~~ ~~u2 dx = sin 􀀀1( ~~u~~a ) + c R a~~2~~u~~2~~ dx = ~~1~~a tan 􀀀 1 ~~u~~a + c R ln (x)dx = (xln(x)) 􀀀 x + c  \nU-Substitution  \nLet u = f (x) (can be more than one variable) .  \nDetermine: du = fd(xx) dx and solve fordx.  \nThen, if a de􀀌nite integral, substitute the bounds for u = f (x) at each bounds Solve the integral using u.  \nIntegration by Parts  \nR udv = uv 􀀀 R vdu  \nFns and Identities  \nsin(cos 􀀀1(x)) = p1 􀀀 x2  \ncos(sin 􀀀1(x)) = p 1 􀀀 x2  \nsec(tan 􀀀1(x)) = p 1 + x2 tan(sec 􀀀1 (x))  \n= ( px2 􀀀 1 if x 􀀕 1)  \n= (􀀀px2 􀀀 1 if x 􀀔 􀀀1)  sinh 􀀀1(x) = ln x + px2 + 1  \nsinh 􀀀1(x) = ln x + px2 􀀀 1; x 􀀕 􀀀1 tanh 􀀀1(x) = ~~1~~2 ln x + ~~1~~1~~x~~x ; 1 \u003C x \u003C 􀀀1 sech 􀀀1(x) = ln[ ~~ ~~1+~~ ~~p1x~~ ~~􀀀~~ ~~x2 ] ; 0 \u003C x 􀀔 􀀀1 sinh(x) = ~~e~~x~~ ~~􀀀2~~e ~~􀀀~~ ~~x  \ncosh(x) = ~~e~~x~~ +~~2~~e ~~􀀀~~ ~~x  \nTrig Identities  \nsin2 (x) + cos2 (x) = 1  \n1 + tan2 (x) = sec2 (x)  \n1 + cot2 (x) = csc2 (x)  \nsin(x 􀀆 y) = sin(x) cos(y) 􀀆 cos(x) sin(y)  \ncos(x 􀀆 y) = cos(x) cos(y) 􀀆 sin(x) sin(y)  \ntan(x 􀀆 y) = 1ta􀀇nta(xn)(􀀆x)tatnan(y(y))  \nsin(2x) = 2 sin(x) cos(x)  \ncos(2x) = cos2 (x) 􀀀 sin2 (x) cosh(n2 x) 􀀀 sinh2 x = 1  \n1 + tan2 (x) = sec2 (x)  \n1 + cot2 (x) = csc2 (x) sin2 (x) = 1~~ ~~􀀀~~ ~~co2s(2x) cos2 (x) = 1+cos2(2x) tan2 (x) = 11coscos((2x2x)) sin(􀀀x) = 􀀀 sin(x) cos(􀀀x) = cos(x) tan(􀀀x) = 􀀀 tan(x)  \nCalculus 3 Concepts Cartesian coords in 3D  \ngiven two points:  \n(x1 ; y1 ; z1 ) and (x2 ; y2 ; z2 ), Distance between them:  \np(x1 􀀀 x2 )2 + (y1 􀀀 y2 )2 + (z1 􀀀 z2 )2 Midpoint:  \n( x~~1 ~~+2x~~2~~ ; y~~1 ~~+2y~~2~~ ; z~~1 ~~+2z~~2~~ )  \nSphere with center (h,k,l) and radius r:(x 􀀀 h)2 + (y 􀀀 k)2 + (z 􀀀 l)2 = r2  \nVectors  \nVector: ~u  \nUnit Vector: u^    \nMagnitude: jj~ujj = qu21 + u22 + u23  \nUnit Vector: u^ = jj jj  \nDot Product  \n~u 􀀁 ~v  \nProduces a Scalar  \n(Geometrically, the dot product is av~ector projection)  \nu~ = \u003C u1 ; u2 ; u3 > v~ = \u003C v; v2 ; v3 >  \nu 􀀁 ~v = 0 means the two vectors are Perpendicular 􀀒 is the angle between t~hem.  \nu~ 􀀁 ~v = jj~u jj jj~v jj cos(􀀒) u 􀀁 ~v = u1 v1 + u2 v2 + u3 v3  \nNOTE:  \nu^ 􀀁 v^ = cos(􀀒) jj~u jj 2 = ~u 􀀁 ~u~u 􀀁 ~v = 0 when ?  \nAngle Between ~u and ~v:  \n􀀒 = cos 􀀀1 ( ~~ ~~jj~~ ~~~~u~~j~j~~ ~~􀀁~~ ~~~~~v~~jj~~ ~~~v~~ ~~jj~~ ~~ )  \nProjection of ~u onto ~v:  \npr~v ~u = ( ~~ ~~jj􀀁~~ ~~~~2~~ )~v  \nCross Product  \n~u 􀀂 ~v  \nProduces a Vector  \n(Geometrically, the cross product is the area of a paralellogram with sides jj~ujja~nd jj~v jj)  \nu~ = \u003C u1 ; u2 ; u3 > v = \u003C v1 ; v2 ; v3 >  \n~ 􀀌 ^i ^j ^k 􀀌~u 􀀂 v = 􀀌 u1 u2 u3 􀀌  \n􀀌 v 1 v2 v3 􀀌  \n~u 􀀂 ~v = ~0 means the vectors are paralell  \nLines and Planes Equation of a Plane  \n(x0 ; y0 ; z0 ) is a point on the plane and \u003C A; B; C > is a normal vector  \nA (x 􀀀 x0 ) + B (y 􀀀 y0 ) + C (z 􀀀 z0 ) = 0 \u003C A; B; C > 􀀁 \u003C x􀀀x0 ; y􀀀y0 ; z􀀀z0 >= 0 Ax + By + Cz = D where  \nD = Ax0 + By0 + Cz","cbCailZ99EHc2kcj","https://ap.wps.com/l/cbCailZ99EHc2kcj","pdf",345826,"English","# Derivatives\n## Integrals\n# Techniques\n## U-Substitution\n## Integration by Parts\n# Functions and Identities\n## Trig Identities\n## Hyperbolic Identities\n# Calculus 3 Concepts\n## 3D Coordinate Geometry\n## Vectors and Products\n# Coordinate System Conversion\n# Surfaces and Partial Derivatives","[{\"question\":\"What derivative rules are included in the cheat sheet?\",\"answer\":\"It lists derivatives for inverse trig functions, hyperbolic functions (sinh, cosh, tanh, coth, sech, csch), and key trigonometric derivatives such as sin(x), cos(x), tan(x), sec(x), and csc(x).\"},{\"question\":\"Which integration techniques and formulas does the document provide?\",\"answer\":\"It provides a set of standard integral forms (including inverse trig-related integrals and logarithmic/arctangent forms) and outlines u-substitution with u=f(x) and integration by parts using ∫u dv = uv − ∫v du.\"},{\"question\":\"What 3D geometry and vector topics are covered?\",\"answer\":\"It summarizes 3D distance and midpoint formulas, sphere equation, plane and line equations, dot and cross products (including geometric interpretations), coordinate conversions between rectangular/cylindrical/spherical, common surface equations, and the definition of partial derivatives.\"}]","Calculus 3 Cheat Sheet - Key Derivatives, Integrals, and 3D Concepts | PDF",1791927320]