[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-571649-105":53,"doc-detail-571649-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","calculus-3-cheat-sheet-reduced-math-formulas-and-key-concepts","Calculus 3 Cheat Sheet - Reduced - Math formulas and key concepts","","Concise calculus 3 reference notes covering derivatives, integrals, and standard calculus techniques. The content lists derivative rules for trigonometric and hyperbolic functions, inverse-function identities, and integration formulas including logarithmic, exponential, and trigonometric integrals. It also summarizes substitution (u-substitution) and integration by parts, followed by trigonometric and inverse-function identities. Additional sections provide 3D analytic geometry essentials, including vectors, dot and cross products, plane and line equations, coordinate-system conversions, quadric surfaces, and partial-derivative/notation guidance.",{"@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/calculus-3-cheat-sheet-reduced-math-formulas-and-key-concepts/571649/",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-reduced-math-formulas-and-key-concepts/571649.png","ImageObject",442,249,{"name":88,"@type":89},"Rainbow Cat","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-25","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":101},"InteractionCounter",{"@type":100},"ViewAction",8,{"@type":103,"mainEntity":104},"FAQPage",[105,111,115],{"name":106,"@type":107,"acceptedAnswer":108},"What derivative formulas are provided for trigonometric and hyperbolic functions?","Question",{"text":109,"@type":110},"The cheat sheet lists derivatives for sin, cos, tan, cot, sec, and csc, along with corresponding hyperbolic-function derivatives such as sinh, cosh, tanh, coth, sech, and csch.","Answer",{"name":112,"@type":107,"acceptedAnswer":113},"Which integration methods are included in these notes?",{"text":114,"@type":110},"It includes basic integral rules for common forms and describes u-substitution steps (set u=f(x), change du, substitute bounds) and the integration by parts formula ∫u dv = uv − ∫v du.",{"name":116,"@type":107,"acceptedAnswer":117},"What 3D geometry and vector topics are covered?",{"text":118,"@type":110},"It covers Cartesian distance and midpoint, sphere equation, vector magnitude, dot product properties (including perpendicular condition), cross product meaning, plane and line equations, and conversion between cylindrical, spherical, and rectangular coordinates.","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},571649,1792002475,{"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":9,"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":4},962090769181,"https://ap-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8","Derivatives  \nDxex = 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)  \nDx csc(x) = − csc(x)cot(x) Dx sin − 1 = p1x2 , x 2 [−1 , 1] Dx cos − 1 = px2 , x 2 [−1 , 1] Dx tan − 1 = 1+~~1~~x~~2~~ , ~~⇡~~ 􀀤 x 􀀤 ~~⇡~~2 Dx sec − 1 = |~~ ~~x|~~ ~~p~~1~~x2~~ ~~−~~ ~~1 , |x| > 1 Dx sinh(x) = cosh(x)  \nDx cosh(x) =  \nDx tanh(x) =  \nDx coth(x) =  \nDx sech(x) =  \nDxcsch(x) =  \n− sinh(x)  \nsech2 (x)  \n−csch2 (x)−sech(x)tanh(x)−csch(x)coth(x)  \nDx sinh − 1 = px~~1~~2+1 Dx cosh − 1 = p2~~1~~−~~ ~~1 , x > 1 Dx tanh − 1 = 1x~~2~~ − 1 \u003C x \u003C 1 Dxsech − 1 = xp1~~1~~−~~ ~~x2 , 0 \u003C x \u003C 1 Dx ln(x) = ~~1~~x  \nIntegrals  \nR ~~1~~x dx = ln |x| + c R exdx = ex + c R axdx = ~~1~~lnaax + c  \nR eaxdx = ~~1~~a eax + c R 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 | cosh(x)| + 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 | cos(x)| + c R cot(x)dx = ln | sin(x)| + 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 = fdx(x) dx and solve fordx.  \nThen, if a definite 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)) = p1 − x2  \nsec(tan−1(x)) = p1 + 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 + ~~x~~x , 1 \u003C x \u003C −1 sech−1(x) = ln[ ~~ ~~1+px1~~ ~~−~~ ~~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) cos(x ± y) = cos(x)cos(y) ± sin(x)sin(y) tan(x ± y) = 1ta⌥nta(xn()±x)tatn(any(y))  \nsin(2x) = 2 sin(x)cos(x)  \ncos(2x) = cos2 (x) − sin2 (x) cosh(n2x) − sinh2 x = 1  \n1 + tan2 (x) = sec2 (x)  \n1 + cot2 (x) = csc2 (x) sin2 (x) = 1~~ ~~−~~ ~~cos2(2x) cos2 (x) = 1+cos2(2x) tan2 (x) = 11cocos(2s(2xx)) 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:   \nUnit Vector: uˆ    \nMagnitude: || || = qu21 + u22 + u23  \nUnit Vector: uˆ = ||~~ ~~||  \nDot Product  \n ·   \nProduces a Scalar  \n(Geometrically, the dot product is a vector projection)  \n = \u003C u1 , u2 , u3 >  \n = \u003C v1~, v2 , v3 >  ·  = 0 means the two vectors are Perpendicular ✓ is the angle between them.  \n ·  = || || || || cos(✓)  \n ·  = u1v1 + u2v2 + u3v3  \nNOTE:  \nuˆ · vˆ = cos(✓)|| || 2 =  ·   ·  = 0 when ?  \nAngle Between  and :✓ = cos−1 ( ~~ ~~||~~ ~~~~u~~|~|~~ ~~·~~ ~~||~~ ~~ )  \nProjection~o  onto : pr = ( ~~ ~~||·~~2~~ )  \nCross Product  \n ⇥   \nProduces a Vector  \n(Geometrically, the cross product is the area of a paralellogram with sides || || and || ||)  \n = \u003C u1 , u2 , u3 >  \n = \u003C v1 , v2 , v3 >  \n$ ˆi ˆj ˆk $  \n ⇥  = $u1 u2 u3 $$v1 v2 v3 $  \n ⇥  = ~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 + Cz0  \nEquation of a line  \nA line requires a Direction Vector  = \u003C u1 , u2 , u3 > and a point (x1 , y1 , z1)  \nthen,  \na parameterization of a line could be: x = u1t + x1 ","cbCaiqNxkPSNErM3","https://ap.wps.com/l/cbCaiqNxkPSNErM3","pdf",532587,"English","# Derivatives\n## Trigonometric and hyperbolic derivative rules\n## Derivatives and inverse-function identities\n# Integrals\n## Basic integral formulas\n## U-Substitution\n## Integration by Parts\n# Functions and Identities\n## Trigonometric identities\n## Inverse function identities\n# Calculus 3 Concepts\n## 3D coordinate geometry and distance/ midpoint\n## Vectors: dot, cross, projections\n## Plane and line equations\n## Coordinate system conversions\n## Surfaces: quadric surfaces\n# Partial Derivatives\n## Definition and notation","[{\"question\":\"What derivative formulas are provided for trigonometric and hyperbolic functions?\",\"answer\":\"The cheat sheet lists derivatives for sin, cos, tan, cot, sec, and csc, along with corresponding hyperbolic-function derivatives such as sinh, cosh, tanh, coth, sech, and csch.\"},{\"question\":\"Which integration methods are included in these notes?\",\"answer\":\"It includes basic integral rules for common forms and describes u-substitution steps (set u=f(x), change du, substitute bounds) and the integration by parts formula ∫u dv = uv − ∫v du.\"},{\"question\":\"What 3D geometry and vector topics are covered?\",\"answer\":\"It covers Cartesian distance and midpoint, sphere equation, vector magnitude, dot product properties (including perpendicular condition), cross product meaning, plane and line equations, and conversion between cylindrical, spherical, and rectangular coordinates.\"}]","Calculus 3 Cheat Sheet - Reduced - Math formulas and key concepts | PDF",1791922954]