[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572657-105":53,"doc-detail-572657-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","differentiation-cheat-sheet-ch12-first-principle-tangents-and-normals","Differentiation Cheat Sheet - Ch.12 - First Principle, Tangents and Normals","","Differentiation is presented as a method to compute the gradient (slope) of a function at different points and to analyze how one variable changes with respect to another. The notes introduce derivative notation, relate gradients to tangents, and derive the derivative from first principles using limits. Worked examples show differentiating power forms and quadratics, finding gradients at specific points, and determining equations of tangents and normals. Increasing and decreasing criteria via the derivative sign are also included.",{"@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":46,"@type":70,"position":76},"https://docshare.wps.com/template/paper-templates/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/differentiation-cheat-sheet-ch12-first-principle-tangents-and-normals/572657/",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/differentiation-cheat-sheet-ch12-first-principle-tangents-and-normals/572657.png","ImageObject",442,249,{"name":88,"@type":89},"I feel Different","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-15","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":9},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"How is differentiation used to find the gradient of a curve?","Question",{"text":108,"@type":109},"The gradient of a curve at a point equals the gradient of the tangent at that point. Differentiation provides that exact gradient by evaluating the derivative at the chosen coordinate.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"What does “differentiating from the first principle” mean?",{"text":113,"@type":109},"It means using the definition of the derivative based on limits of the difference quotient as h tends to 0. The notes show substituting values into this limit to obtain the derivative.",{"name":115,"@type":106,"acceptedAnswer":116},"How do you find the equations of a tangent and a normal to a curve at a point?",{"text":117,"@type":109},"Compute the derivative to get the tangent gradient at the point, then use the tangent line equation with point and slope. The normal is perpendicular to the tangent, so its gradient is the negative reciprocal, and it uses the same point with that slope.","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},572657,1792053788,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":45,"category_name":46,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":9,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":9,"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":4},962093541761,"https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0","Differentiation Cheat Sheet  \nDifferentiation is a process that helps us to calculate gradient or slope of a function at different points. It also help us to identify change in one variable with respect to another variable. You will learn real life application of differentiation in this topic  \nDerivative Notation:  \n!!\"\\# or 􀀢$ (􀀤) represents derivative of 􀀦 = 􀀢(􀀤) with respect to x  \nGradient of curves:  \nUnlike straight lines, the gradient of a curve changes constantly. The gradient of a curve at any given point is the same as calculating gradient of the tangent at that given point. You can find exact gradient of a curve at any given point using derivatives.  \nFinding derivative:  \nIn this section you will learn how to find derivative of a function (i.e. exact gradient of a curve or function at a given point)  \nConsider the following figure for a curve 􀀦 = 􀀢 (􀀤)  \nAs point 􀁓 moves closer to point 􀁆 , the gradient of chord 􀁆􀁓 gets closer to the gradient of tangent to the curve at 􀁆 .  \nThe coordinate of 􀁓 is (􀀣$, 􀀡 (􀀣$)) and 􀁕 is (􀀣$ + ℎ, 􀀡 (􀀣$ + ℎ)) . So, the gradient of AB is-(($0\"\")*-(($) . As ℎ gets smaller, gradient of 􀁓􀁕 gets closer to gradient of curve at 􀁓 .  \nHence,  \nThe gradient function or the derivative of a curve 􀀷 = 􀀢 (􀀤) is given by  \n􀁎! (􀁏)  \n= 􀁔􀁕􀁖  \n􀀱→􀀲  \n􀀳(􀀴0􀀱)*􀀳(􀀴)􀀱  \nlim  \n\"→$  \nmeans the limit as ℎ tends to 0.  \nThis rule is called differentiating from the first principle  \nExample 1: 􀀢 (􀀤) = 􀀤 % a. Show that 􀀢 $ (􀀤) = lim (2􀀤 + ℎ) b. Hence deduce that 􀀢 $ (􀀤) = 2􀀤  \n&→(  \na. Use the definition of derivative  \n􀀡 ! (􀀣) = lim 􀀡 (􀀣 + ℎ) − 􀀡(􀀣)  \n\"→$ ℎ  \nSo 􀀡(􀀣) = 􀀣 % implies 􀀡 (􀀣 + ℎ) = (􀀣 + ℎ)%,  \nSubstitute 􀀡 (􀀣 + ℎ) and 􀀡(􀀣) into the above definition of  \nderivative  \n(􀀣 + ℎ)% − 􀀣 %􀀡 ! (􀀣) = lim ~~ ~~  \n\"→$ ℎ  \n􀀡 ! (􀀣)  \n= lim  \n\"→$  \n􀀣 % + 2􀀣ℎ + ℎ% − 􀀣 %  \nℎ  \nExpand the bracket  \n(􀀢 + ℎ)% = (􀀢 + ℎ)(􀀢 += 􀀢 % + 2􀀢ℎ +  \nFactorise the numerator ↓  \n􀀡 ! (􀀣) = lim  2􀀣ℎ + ℎ% ⟹ 􀀡 ! (􀀣)  \n\"→$ ℎ  \n􀀡 ! (􀀣) = lim(2􀀣 + ℎ)  \n\"→$  \nDifferentiating 􀀡􀀡  \n= lim  \n\"→$  \nℎ(2􀀣 + ℎ)  \nℎ  \nℎ)  \nℎ%  \nb.  \n􀀢 ! (􀀤) = 2􀀤  \nFrom part a, we know that 􀀢 ! (􀀤) = lim (2􀀤 + ℎ)  \n\"→$ Apply the limits  \nAs h→0, 2􀀤 + ℎ → 2􀀤  \nh→0 means as h tends to 0 or h approaches to 0, 2􀀤 + ℎ approaches 2􀀤  \nHence,  \n􀀢 ! (􀀤) = lim (2􀀤 + ℎ) = 2􀀤  \n\"→$  \nIn this chapter you will only learn how to differentiate functions of the form 􀀣&, where 􀀯 is any number. Use the following results to differentiate functions.  \n􀀰 = 􀀡 (􀀣) = 􀀣& then ''() = 􀀡 ! (􀀣) = 􀀯􀀣&*+ where 􀀯 is any real number  \n􀀰 = 􀀡 (􀀣) = 􀀱􀀣& then ''() = 􀀡 ! (􀀣) = 􀀱􀀯􀀣&*+ where 􀀯 is any real number and 􀀱 is a constant  \nHere is an example for you to understand how to apply the results Example 2: Find the derivative, 􀀡! (􀀣) when 􀀡(􀀣) equals:  \na. 􀀣 , b. 10􀀣*+  \na. 􀀡 (􀀣) = 􀀣 ,  \nYou are supposed to find derivative of 􀀡(􀀣) = 􀀣 ,, use the result for 􀀣&  \nSo 􀀡! (􀀣) = 6􀀣,*+ = 6􀀣-  \nb. 􀀡 (􀀣) = 10􀀣*+  \nUse the result for 􀀱􀀣&  \n􀀡 ! (􀀣) = 10 × (−1) × 􀀣*+*+ = −10 􀀣 *%  \n􀀡 ! (􀀣) = −10 × )+! = − $!~~ ~~ As 􀀣 *% = )+!  \nDifferentiating quadratics:  \nA quadratic function or a curve is given by 􀀷 = 􀀸􀀤% + 􀀹􀀤 + 􀀺, where 􀀸, 􀀹 and 􀀺 are constants. The derivative of 􀀷 is &&'( = 2􀀸􀀤 + 􀀹  \nNote: Differentiate each term one at a time  \nDerivative of only a constant term is always 0. So if 􀀷 = 2 then &&'( = 0  \nExample 3: Find the gradient of the curve with equation 􀀷 = 2􀀤% − 􀀤 − 1 at the point (2, 5)  \nAs explained in the gradient of curves section, finding the gradient of a curve at a point is same as calculating derivative of the curve at that point.  \nSo first, you will have to calculate the gradient of 􀀷 = 2􀀤% − 􀀤 − 1  \n&&'( = 2 × 2 × 􀀤(%*+) − 1 × 􀀤(+*+) − 0 Since 1 is a constant, derivative of a constant will be 0 &&'( = 4􀀤 − 􀀤 $ = 4􀀤 − 1 As 􀀤 $ = 1  \nNow to find the derivative of 􀀷 at point(2, 5), you need to substitute 􀀤 = 2 in the derivative function 􀁂􀁂􀀷􀀤 = 􀀢 ! (2) = 4 (2) − 1 = 8 − 1 = 7  \nHence, the gradient of the curve with equation 􀀷 = 2􀀤% − 􀀤 − 1 at the point (2, 5) is 7.  \nGradient, tangents and normal  ","cbCaih4IUxvzPKZh","https://ap.wps.com/l/cbCaih4IUxvzPKZh","pdf",566698,"English","# Differentiation Cheat Sheet\n## Derivative notation and gradient of curves\n## Derivative from first principles\n## Differentiating power functions and examples\n## Differentiating quadratics and gradient at a point\n## Gradient, tangents and normal\n## Increasing and decreasing functions","[{\"question\":\"How is differentiation used to find the gradient of a curve?\",\"answer\":\"The gradient of a curve at a point equals the gradient of the tangent at that point. Differentiation provides that exact gradient by evaluating the derivative at the chosen coordinate.\"},{\"question\":\"What does “differentiating from the first principle” mean?\",\"answer\":\"It means using the definition of the derivative based on limits of the difference quotient as h tends to 0. The notes show substituting values into this limit to obtain the derivative.\"},{\"question\":\"How do you find the equations of a tangent and a normal to a curve at a point?\",\"answer\":\"Compute the derivative to get the tangent gradient at the point, then use the tangent line equation with point and slope. The normal is perpendicular to the tangent, so its gradient is the negative reciprocal, and it uses the same point with that slope.\"}]","Differentiation Cheat Sheet - Ch.12 - First Principle, Tangents and Normals | PDF",1791928742]