[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-571483-105":53,"doc-detail-571483-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","differentiation-cheat-sheet-learn-rules-and-methods","Differentiation Cheat Sheet - learn rules and methods","","Differentiation Cheat Sheet consolidates essential calculus techniques for computing derivatives of more complex functions. It covers foundational proofs from first principles, including derivatives of core exponential and logarithmic forms, as well as differentiation of exponentials and logarithms. The sheet then explains the chain rule, product rule, quotient rule, and how they apply to composite expressions. It further provides results for differentiating key trigonometric functions and demonstrates the methods through worked examples, concluding with parametric and implicit differentiation.",{"@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/differentiation-cheat-sheet-learn-rules-and-methods/571483/",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-learn-rules-and-methods/571483.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":101},"InteractionCounter",{"@type":100},"ViewAction",6,{"@type":103,"mainEntity":104},"FAQPage",[105,111,115],{"name":106,"@type":107,"acceptedAnswer":108},"What are the two main results the cheat sheet requires proving from first principles?","Question",{"text":109,"@type":110},"The sheet asks you to know two derivative results and prove them from first principles: [sin x] differentiation is indicated as a worked item, and it also specifically targets the proofs involving the expressions shown as 􀯗􀯗􀯫 [􀝏􀝅􀝊􀝔] = 􀜿􀝋􀝏􀝔 and 􀯗􀯗􀯫 [􀜿􀝋􀝏􀝔] = −􀝏􀝅􀝊􀝔.","Answer",{"name":112,"@type":107,"acceptedAnswer":113},"How does the chain rule help when differentiating composite functions?",{"text":114,"@type":110},"The chain rule is described as a method for differentiating composite expressions where one function is inside another. It provides formulas for cases like y = [f(x)]^n and y = f(g(x)], and shows how to apply them in examples.",{"name":116,"@type":107,"acceptedAnswer":117},"Which rules are included for differentiating products and quotients of functions?",{"text":118,"@type":110},"The sheet includes the product rule for y = uv and gives a convenient derivative form for u v. It also includes the quotient rule for y = u/v and notes that the quotient rule can be replaced by using the product rule after rewriting with negative powers.","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},571483,1792022767,{"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":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},687212989558,"https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","Differentiation Cheat Sheet  \nPreviously in Pure Year 1, you only learnt how to differentiate simple expressions, such as 2􀝔2. In this chapter, we will learn more rules and methods that will enable us to differentiate much more complicated functions.  \nDifferentiating 􀜛􀜑􀜖􀜠 and 􀜋􀜗􀜛􀜠  \nYou need to be know the following two results, and be able to prove 􀯗􀯗􀯫 [􀝏􀝅􀝊􀝔] = 􀜿􀝋􀝏􀝔 and 􀯗􀯗􀯫 [􀜿􀝋􀝏􀝔] = −􀝏􀝅􀝊􀝔 from first principles.  \n▪ ~~ ~~􀯗􀯗􀯫 [sin 􀝇􀝔] = 􀝇􀜿􀝋􀝏 􀝇􀝔  \n▪ ~~ ~~􀯗􀯗􀯫 [cos 􀝇􀝔] = −􀝇􀝏􀝅􀝊 􀝇􀝔  \nExample 1: Prove, from first principles, that the derivative of 􀝏􀝅􀝊􀝔 is 􀜿􀝋􀝏􀝔 . You may assume that as ℎ → 0,  \n􀯦􀯜􀯡ℎℎ → 1 and 􀯖􀯢􀯦ℎℎ−1 → 0.  \nWhenever you want to prove a result from first principles, we must use the following definition of the derivative:  \n􀝂′(􀝔) = lℎi [􀝂(􀝔~~ ~~+~~ ~~ℎℎ)~~ ~~−~~ ~~􀝂(􀝔)]  \nLetting 􀝂(􀝔) = 􀝏􀝅􀝊􀝔 and using the above definition:  \nUsing the addition formula, 􀝏􀝅􀝊(􀜣 + 􀜤)  \n􀝂′(􀝔) = lℎi [􀝂(􀝔~~ ~~+~~ ~~ℎℎ)~~ ~~−~~ ~~􀝂(􀝔)] = lℎi [􀝏􀝅􀝊(􀝔~~ ~~+~~ ~~ℎℎ)~~ ~~−~~ ~~􀝏􀝅􀝊􀝔] = lℎi [􀝏􀝅􀝊􀝔􀜿􀝋􀝏ℎ + 􀜿􀝋􀝏ℎ􀝔􀝏􀝅􀝊ℎ − 􀝏􀝅􀝊􀝔]= lℎi [􀝏􀝅􀝊􀝔(􀜿􀝋􀝏ℎ~~ ~~−~~ ~~1ℎ)~~ ~~+~~ ~~􀜿􀝋􀝏􀝔􀝏􀝅􀝊ℎ] = lℎi [􀝏􀝅􀝊􀝔(􀜿􀝋􀝏ℎℎ~~ ~~−~~ ~~1) + 􀜿􀝋􀝏􀝔ℎ(sinh)]  \n= lim [􀝏􀝅􀝊􀝔 (0) + 􀜿􀝋􀝏􀝔(1)] = lim [􀜿􀝋􀝏􀝔] = 􀜿􀝋􀝏􀝔  \nℎ→0 ℎ→0  \nFactorising out 􀝏􀝅􀝊􀝔 in the numerator, and then splitting up the fraction  \nUsing the assumptions given to us in the question  \nThe proof for the derivative of 􀜿􀝋􀝏􀝔 is very similar. We would start by instead letting 􀝂(􀝔) = 􀜿􀝋􀝏􀝔 .  \nDifferentiating exponentials and logarithms  \nYou should also remember the following results:  \n▪ ~~ ~~􀯗􀯗􀯫 [􀝁 􀯞􀯫 ] = 􀝇􀝁 􀯞􀯫  \n▪ ~~ ~~􀯗􀯗􀯫 [􀜽􀯞􀯫 ] = 􀜽􀯞􀯫 (􀝇 ln 􀜽)  \n▪ ~~ ~~􀯗􀯗􀯫 [ln 􀝔] = 1􀯫  \nYou need to know how to prove the second result.  \nExample 2: Show that the derivative of 􀜽􀯫 is 􀜽􀯫 (ln 􀜽) .  \nWe start by letting 􀝕 = 􀜽􀯫 . Now using the following property of exponentials: 􀝁 􀯟􀯡􀯙(􀯫) = 􀝂 (􀝔),  \n⇒ 􀝕 = 􀝁 ln (􀯔􀳣 ) = 􀝁 􀯫(􀯟􀯡􀯔)  \n∴ 􀯗􀯬􀯗􀯫 = (ln 􀜽) × 􀝁 􀯫(􀯟􀯡􀯔) ~~ ~~ Using the result: 􀯗􀯗􀯫 [􀝁 􀯞􀯫 ] = 􀝇􀝁 􀯞􀯫 with 􀝇 = ln 􀜽  \nBut since we said in the first line that 􀝕 = 􀝁 􀯫(􀯟􀯡􀯔), we have that  \n􀯗􀯬􀯗􀯫 = (ln 􀜽) × 􀝕 = 􀯗􀯗􀯬􀯫 = (ln 􀜽) × 􀜽􀯫 as required.  \nThe chain rule  \nThe chain rule is a powerful method used to differentiate composite functions (i. e. expressions where one function is contained in another function) . An example of such a function would be 􀝁 􀯫2−3􀯫+1, where the function 􀝔2 − 3􀝔 + 1 is contained inside the function 􀝁 􀯫 . This rule allows us to differentiate seemingly complex expressions with ease and plays a pivotal role in this chapter.  \nThe chain rule is:  \n▪ ~~ ~~􀯗􀯗􀯬􀯫 = 􀯗􀯗􀯬􀯨 × 􀯗􀯗􀯨􀯫 , where 􀝕 is a function of 􀝑 and 􀝑 is another function of 􀝔 . We can write this in function notation, which tends to be easier to use in application:  \n▪ If 􀝕 = [􀝂 (􀝔)]􀯡 , then 􀯗􀯬􀯗􀯫 = 􀝊 [􀝂 (􀝔)]􀯡−1 × 􀝂′(􀝔) [1]  \n▪ If 􀝕 = 􀝂 [􀝃 (􀝔)] then 􀯗􀯬􀯗􀯫 = 􀝂′ [􀝃 (􀝔)] × 􀝃 ′(􀝔) [2]  \nWhen differentiating functions that are not of the form 􀝕 = 􀝂 (􀝔), the following case of the chain rule is useful:  \n▪  \n􀯗􀯬  \n􀯗􀯫  \n=  \n1  \n􀳏􀳣  \n􀳏􀳤  \nHere are two examples showing the chain rule in action:  \n\n| Example 3: Differentiate (􀝏􀝅􀝊􀝔 + 􀜿􀝋􀝏􀝔)5 |\n| --- |\n| This is of the form [1], where 􀝂 (􀝔) = 􀝏􀝅􀝊􀝔 + 􀜿􀝋􀝏􀝔 and 􀝊 = 5.\u003Cbr>⇒ 􀝀􀝀􀝕􀝔 = 􀝊 [􀝂(􀝔)]􀯡−1 × 􀝂′(􀝔)\u003Cbr>= 5[􀝏􀝅􀝊􀝔 + 􀜿􀝋􀝏􀝔]4 × (􀜿􀝋􀝏􀝔 − 􀝏􀝅􀝊􀝔)\u003Cbr>= 5[􀝏􀝅􀝊􀝔 + 􀜿􀝋􀝏􀝔]4 (􀜿􀝋􀝏􀝔 − 􀝏􀝅􀝊􀝔) |\n\nThe product rule  \n\n| Example 4: Differentiate 􀝁 􀯫2−4􀯫+2 |\n| --- |\n| This is of the form [2], where\u003Cbr>􀝃 (􀝔) = 􀝔2 − 4􀝔 + 2 and 􀝂(􀝔) = 􀝁 􀯫 .⇒ 􀝂′(􀝔) = 􀝁 􀯫 and 􀝃′(􀝔) = 2􀝔 − 4 ∴ 􀝀􀝀􀝕􀝔 = 􀝂′ [􀝃 (􀝔)] × 􀝃 ′(􀝔)\u003Cbr>= 􀝂′ [􀝔2 − 4􀝔 + 2] × (2􀝔 − 4)\u003Cbr>= 􀝁 􀯫2−4􀯫+2 × (2􀝔 − 4) = (2􀝔 − 4)􀝁 􀯫2−4􀯫+2 |\n\nWhen we want to differentiate an expression that is a product of two functions, we can use the product rule. The product rule is:  \n▪ If 􀝕 = 􀝑􀝒, then 􀯗􀯗􀯬􀯫 = 􀝑 􀯗􀯗􀯩􀯫 + 􀝒 􀯗􀯗􀯨􀯫 where 􀝑 and 􀝒 are both functions of 􀝔  \nA more convenient way to write this is  \n􀯗􀝀􀯬􀝔 = 􀝑􀝒 ′ + 􀝒􀝑′  \nwhere 􀝑′ and 􀝒′ are derivatives of 􀝑 and 􀝒 with respect to 􀝔 .  \nThe quotient rule  \nWhen we want to differentiate an expression that is the quotient of two functions, we can use the quotient rule. The quotient rule is:  \n▪ If 􀝕 = 􀝑􀝒, then 􀯗􀯗􀯬􀯫 = 􀯩􀳏~~􀳏","cbCairNnBZGrDijV","https://ap.wps.com/l/cbCairNnBZGrDijV","pdf",743554,"English","# Differentiating from first principles\n## Derivatives of key exponential/logarithmic results\n# Differentiating exponentials and logarithms\n# The chain rule\n## Function-notation forms\n## Worked examples\n# The product rule\n# The quotient rule\n# Differentiating trigonometric functions\n## Strategy using transformations\n# Parametric differentiation\n# Implicit differentiation","[{\"question\":\"What are the two main results the cheat sheet requires proving from first principles?\",\"answer\":\"The sheet asks you to know two derivative results and prove them from first principles: [sin x] differentiation is indicated as a worked item, and it also specifically targets the proofs involving the expressions shown as 􀯗􀯗􀯫 [􀝏􀝅􀝊􀝔] = 􀜿􀝋􀝏􀝔 and 􀯗􀯗􀯫 [􀜿􀝋􀝏􀝔] = −􀝏􀝅􀝊􀝔.\"},{\"question\":\"How does the chain rule help when differentiating composite functions?\",\"answer\":\"The chain rule is described as a method for differentiating composite expressions where one function is inside another. It provides formulas for cases like y = [f(x)]^n and y = f(g(x)], and shows how to apply them in examples.\"},{\"question\":\"Which rules are included for differentiating products and quotients of functions?\",\"answer\":\"The sheet includes the product rule for y = uv and gives a convenient derivative form for u v. It also includes the quotient rule for y = u/v and notes that the quotient rule can be replaced by using the product rule after rewriting with negative powers.\"}]","Differentiation Cheat Sheet - learn rules and methods | PDF",1791919930]