[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-571564-105":53,"doc-detail-571564-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-cheat-sheet-derivatives","Calculus Cheat Sheet - Derivatives","","A concise calculus derivatives reference covering definition and notation for dy/dx, multiple equivalent derivative forms, and interpretations linking derivatives to tangent slope, instantaneous rate of change, and velocity. Presents core differentiation properties and key rules such as the product, quotient, power, and chain rule, followed by common derivative results for trigonometric and logarithmic/exponential functions. Includes guidance for chain-rule variants, higher-order derivatives, implicit differentiation workflow, and methods for analyzing critical points, increasing/decreasing behavior, concavity, inflection points, and extrema.",{"@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":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-cheat-sheet-derivatives/571564/",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-cheat-sheet-derivatives/571564.png","ImageObject",442,249,{"name":88,"@type":89},"Jordan Avery","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-22","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},"How is the derivative notation defined when y = f(x)?","Question",{"text":109,"@type":110},"If y = f(x), the derivative at x = a represents the slope of the tangent line at x = a, the instantaneous rate of change at x = a, and (when f(x) is position over time) the velocity at x = a.","Answer",{"name":112,"@type":107,"acceptedAnswer":113},"What are the core differentiation rules provided?",{"text":114,"@type":110},"It summarizes linearity, the product rule, the quotient rule, the constant rule ddx(c)=0, the power rule, and the chain rule for compositions f(g(x)).",{"name":116,"@type":107,"acceptedAnswer":117},"How are extrema and concavity analyzed?",{"text":118,"@type":110},"The cheat sheet defines critical points using f '(c)=0 or nonexistence, classifies increasing/decreasing via the sign of f '(x), concavity via the sign of f ''(x), identifies inflection points where concavity changes, and outlines steps and derivative tests for absolute and relative extrema.","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},571564,1792022775,{"code":4,"msg":5,"data":128},{"doc_id":125,"user_id":129,"nickname":88,"user_avatar":130,"doc_module":9,"category_id":8,"category_name":11,"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":73,"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":9},1099523882367,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Calculus Cheat Sheet  \nDerivatives  \nDefinition and Notation  \nIf y = f (x ) then the derivative is defined to bef ′(x ) = lhi ~~ ~~f~~ ~~(x~~ ~~+~~ ~~hh)~~ ~~−~~ ~~f~~ ~~(x~~ ~~) .  \nIf y = f (x ) then all of the following are equivalent notations for the derivative.  \nf ′(x ) = y′= dfdx = dydx = ddx~~ ~~ (f (x ))= Df (x )  \nIf y = f (x )all of the following are equivalent notations for derivative evaluated at x = a . f ′(a ) = y′x =a = dfdx~~ ~~ x =a = dydx~~ ~~ x =a = Df (a )  \nInterpretation of the Derivative  \nIf y = f (x ) then,  \n1. m = f ′(a ) is the slope of the tangent line to y = f (x ) at x = a and the equation of the tangent line at x = a is given by y = f (a )+ f ′(a )(x − a ) .  \n2. f ′(a ) is the instantaneous rate of change of f (x ) at x = a .  \n3. If f (x ) is the position of an object at time x then f ′(a ) is the velocity of the object at x = a .  \nBasic Properties and Formulas  \nIf f (x ) and g (x ) are differentiable functions (the derivative exists), c and n are any real numbers,  \n1. (c f )′= c f ′(x )  \n2. (f ± g )′= f ′(x )± g′(x )  \n3. (f g )′= f ′g + f g′– Product Rule  \n4. ~~ ~~fg ′= ~~ ~~f~~ ~~′g2f~~ ~~g′ – Quotient Rule  \n5. ddx (c ) = 0  \n6. ddx (xn)= nxn −1– Power Rule  \n7. ddx (f (g (x )))= f ′(g (x ))g′(x ) This is the Chain Rule  \nCommon Derivatives  \n d   \ndx  \n d   \ndx  \n d   \ndx  \n d   \ndx  \n d   \ndx  \n(x ) = 1  \n(sin x) = cos x (cos x ) =− sin x (tan x ) = sec2 x (sec x ) = sec x tan x  \n d   \ndx  \n d   \ndx  \n d   \ndx  \n d   \ndx  \n d   \ndx  \n(cscx ) =− cscx cot x (cot x ) =− csc2 x (sin−1 x ) =   1 ( − 1 ) = −1 − x12   \ncos x  \n 1 − x2 (tan −1 x ) = ~~ ~~1~~ +~~1x2~~ ~~  \n d   \ndx  \n d   \ndx  \n d   \ndx  \n d   \ndx  \n d   \ndx  \n(ax)= ax ln (a )  \n(ex)= ex  \n(ln (x ))= ~~ ~~1x~~ ~~ , x > 0  \n(ln x ) = ~~ ~~1x~~ ~~ , x ≠ 0 (loga (x ))= ~~ ~~xl1n~~ ~~a~~ ~~ , x > 0  \nCalculus Cheat Sheet  \nChain Rule Variants  \nThe chain rule applied to some specific functions.  \n1. ddx ( f (x ) n )= n f (x ) n −1 f ′(x )  \n2. ddx (ef (x ) ) = f ′(x )ef (x)  \n3. ddx (ln f (x ) ) = ~~ ~~ff~~ ~~′~~(~~(xx~~)~~)  \n4. ddx (sin f (x ) ) = f ′(x )cos f (x )  \n5.  \n6.  \n7.  \n8.  \nddx (cos f (x ) ) =− f ′(x )sin f (x ) ddx (tan f (x ) ) = f ′(x )sec2 f (x ) ddx (sec [f ( x) ])= f ′( x) sec [f ( x) ]tan [f ( x) ] ddx (tan −1 f (x ) ) = 1~~ ~~+~~ ~~ff′()~~) ~~2~~ ~~  \nHigher Order Derivatives  \nThe Second Derivative is denoted asf ′′(x ) = f (2 ) (x ) = ~~ ~~dd2x~~ ~~ and is defined asf ′(x ) = (f ′(x ))′, i.e. the derivative of the first derivative, f ′(x ) .  \nThe nth Derivative is denoted as  \nf (n ) (x ) = ~~ ~~ddnx~~ ~~ and is defined asf (n ) (x ) = (f (n −1) (x ))′, i.e. the derivative of the (n-1)st derivative, f (n −1) (x ).  \nImplicit Differentiation  \nFind y′if e2 x −9y + x3y 2 = sin (y )+ 11x . Remember y = y (x ) here, so products/quotients ofx andy will use the product/quotient rule and derivatives of y will use the chain rule. The “trick” is to differentiate as normal and every time you differentiate a y you tack on a y′(from the chain rule) . After differentiating solve for y′.  \ne2 x −9y (2 − 9y′)+ 3x2y 2 + 2x3yy′= cos (y )y′+ 11  \n2e2 x−9y − 9y′e2 x−9y + 3x2y 2 + 2x3y y′= cos (y )y′+ 11 ⇒ y′= ~~ ~~21x31~~ ~~−e2~~ ~~xe2y9y~~ ~~−~~−~~3xc2yos~~ ~~)  \n(2x3y − 9e2 x−9y − cos (y ))y′= 11 − 2e2 x−9y − 3x2y2 y y  \nIncreasing/Decreasing – Concave Up/Concave Down  \nCritical Points  \nx = c is a critical point of f (x ) provided either 1. f ′(c ) = 0 or 2. f ′(c ) doesn’t exist.  \nIncreasing/Decreasing  \n1. If f ′(x ) > 0 for all x in an interval I then f (x ) is increasing on the interval I.  \n2. If f ′(x ) \u003C 0 for all x in an interval I then f (x ) is decreasing on the interval I.  \n3. If f ′(x ) = 0 for all x in an interval I then f (x ) is constant on the interval I.  \nConcave Up/Concave Down  \n1. If f ′′(x ) > 0 for all x in an interval I then f (x ) is concave up on the interval I.  \n2. If f ′′(x ) \u003C 0 for all x in an interval I then f (x ) is concave down on the interval I.  \nInflection Points  \nx = c is a","cbCaijzUWSziK0cd","https://ap.wps.com/l/cbCaijzUWSziK0cd","pdf",42987,"English","# Derivatives\n## Definition and Notation\n## Interpretation of the Derivative\n## Basic Properties and Formulas\n## Common Derivatives\n## Chain Rule Variants\n## Higher Order Derivatives\n## Implicit Differentiation\n## Increasing/Decreasing and Concavity\n## Extrema\n### Absolute Extrema\n### Relative (Local) Extrema","[{\"question\":\"How is the derivative notation defined when y = f(x)?\",\"answer\":\"If y = f(x), the derivative at x = a represents the slope of the tangent line at x = a, the instantaneous rate of change at x = a, and (when f(x) is position over time) the velocity at x = a.\"},{\"question\":\"What are the core differentiation rules provided?\",\"answer\":\"It summarizes linearity, the product rule, the quotient rule, the constant rule ddx(c)=0, the power rule, and the chain rule for compositions f(g(x)).\"},{\"question\":\"How are extrema and concavity analyzed?\",\"answer\":\"The cheat sheet defines critical points using f '(c)=0 or nonexistence, classifies increasing/decreasing via the sign of f '(x), concavity via the sign of f ''(x), identifies inflection points where concavity changes, and outlines steps and derivative tests for absolute and relative extrema.\"}]","Calculus Cheat Sheet - Derivatives | PDF",1791922374]