{"id":9290,"date":"2017-03-14t10:31:17","date_gmt":"2017-03-14t17:31:17","guid":{"rendered":"\/\/www.catharsisit.com\/hs\/?p=9290"},"modified":"2017-03-12t10:32:11","modified_gmt":"2017-03-12t17:32:11","slug":"ap-calculus-review-chain-rule","status":"publish","type":"post","link":"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-chain-rule\/","title":{"rendered":"ap calculus review: chain rule"},"content":{"rendered":"<p>of all the derivative rules it seems that the chain rule gets the worst press.  many students dread the rule, think that it&#8217;s too difficult, don&#8217;t fully understand where to apply it, and generally wish that it would go away.  <\/p>\n<p>are you in this group?  <\/p>\n<p>if so then i hope that by the end of this short article, you&#8217;ll gain a better appreciation for the chain rule and how it is used in derivative problems.<\/p>\n<h2>the chain rule<\/h2>\n<p>we use the chain rule to find the derivative of a <strong>composition<\/strong> of functions, that is a function of the form <em>f<\/em>(<em>g<\/em>(<em>x<\/em>)).  <\/p>\n<h3>what is a composition?<\/h3>\n<p>if <em>f<\/em>(<em>x<\/em>) and <em>g<\/em>(<em>x<\/em>) are two functions, then we call <em>f<\/em>(<em>g<\/em>(<em>x<\/em>)) the composition of <em>f<\/em> and <em>g<\/em>.  we might call <em>f<\/em> the &#8220;outside&#8221; function, and <em>g<\/em> the &#8220;inside&#8221; function.<\/p>\n<p>for example, if <em>f<\/em>(<em>x<\/em>) = sin(<em>x<\/em>), and <em>g<\/em>(<em>x<\/em>) = <em>x<\/em><sup>2<\/sup> + 1, then their composition is:<\/p>\n<p><em>f<\/em>(<em>g<\/em>(<em>x<\/em>)) = sin(<em>x<\/em><sup>2<\/sup> + 1).<\/p>\n<h3>decomposition<\/h3>\n<p>to use the chain rule properly, you need to know how to go the other direction.  in other words, you have to learn how to <strong>decompose<\/strong> a composite function into a pair of functions &mdash; the outside and inside functions.<\/p>\n<p>to make it clearer, we often use a different variable name, such as <em>u<\/em>, for the input variable of the outside function.  <\/p>\n<p>for example, if <em>y<\/em> = (4<em>x<\/em> &#8211; 2)<sup>3<\/sup>, then the most natural decomposition would be:<\/p>\n<p><em>(outside function, or <em>f<\/em>(<em>u<\/em>)):<\/em> &nbsp;&nbsp;<em>y<\/em>  = <em>u<\/em><sup>3<\/sup>, and <\/p>\n<p><em>(inside function, or <em>g<\/em>(<em>x<\/em>)):<\/em> &nbsp;&nbsp;<em>u<\/em> = 4<em>x<\/em> &#8211; 2.<\/p>\n<h3>statement of the chain rule<\/h3>\n<p>suppose <em>f<\/em> and <em>g<\/em> are differentiable functions.  then the composite function, <em>f<\/em>(<em>g<\/em>(<em>x<\/em>)) is also differentiable, and<\/p>\n<p><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/chainrule.gif\" alt=\"statement of the chain rule\" width=\"206\" height=\"20\" class=\"aligncenter size-full wp-image-9293\" \/><\/p>\n<p>i think of this as a <strong>3-step process<\/strong>.<\/p>\n<p><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/three-step-process.jpeg\" alt=\"three step process for chain rule\" width=\"361\" height=\"195\" class=\"aligncenter size-full wp-image-9298\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/three-step-process.jpeg 361w, \/\/www.catharsisit.com\/hs\/files\/2017\/02\/three-step-process-300x162.jpeg 300w\" sizes=\"(max-width: 361px) 100vw, 361px\" \/><\/p>\n<ol>\n<li>find the derivative of the outside.<\/li>\n<li>plug in the inside function.<\/li>\n<li>multiply by the derivative of the inside.<\/li>\n<\/ol>\n<p>equivalently, if <em>y<\/em> = <em>f<\/em>(<em>u<\/em>), and <em>u<\/em> = <em>g<\/em>(<em>x<\/em>), then<\/p>\n<p><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/chainrule_leibniz.gif\" alt=\"chain rule in leibniz notation\" width=\"103\" height=\"38\" class=\"aligncenter size-full wp-image-9294\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/chainrule_leibniz.gif 103w, \/\/www.catharsisit.com\/hs\/files\/2017\/02\/chainrule_leibniz-30x11.gif 30w\" sizes=\"(max-width: 103px) 100vw, 103px\" \/><\/p>\n<p>here, the rule has been written in leibniz notation, and clearly expresses the fact that the derivative of a composition is really just the product of the derivatives of the individual functions.  just don&#8217;t forget that step of plugging back in <em>u<\/em> = <em>g<\/em>(<em>x<\/em>).<\/p>\n<h3>example<\/h3>\n<p>find the derivative of <img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/chainrule_example1.gif\" alt=\"y = square root of x^2 + 1\" width=\"98\" height=\"21\" class=\"alignnone size-full wp-image-9295\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/chainrule_example1.gif 98w, \/\/www.catharsisit.com\/hs\/files\/2017\/02\/chainrule_example1-30x6.gif 30w\" sizes=\"(max-width: 98px) 100vw, 98px\" \/>.<\/p>\n<p>first decompose the function.  <\/p>\n<p><em>(outside function):<\/em> &nbsp;&nbsp;<em>f<\/em>(<em>u<\/em>)  = <img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/square_root1.gif\" alt=\"square root of u\" width=\"25\" height=\"19\" class=\"alignnone size-full wp-image-9297\" \/>. <\/p>\n<p><em>(inside function):<\/em> &nbsp;&nbsp;<em>g<\/em>(<em>x<\/em>) = <em>x<\/em><sup>2<\/sup> + 1.<\/p>\n<p>following the three-step process,<\/p>\n<ol>\n<li>first find the derivative of the outside: <img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/derivative_of_square_root.gif\" alt=\"derivative of the outside function\" width=\"191\" height=\"43\" class=\"alignnone size-full wp-image-9299\" \/><\/li>\n<li>next, plug in the inside function: <img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/step2_example_chain_rule.gif\" alt=\"plugging in the inside function\" width=\"73\" height=\"42\" class=\"alignnone size-full wp-image-9300\" \/><\/li>\n<li>finally, multiply by the derivative of the inside: <img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/step3_example_chain_rule.gif\" alt=\"step 3 of the chain rule example\" width=\"120\" height=\"42\" class=\"aligncenter size-full wp-image-9301\" \/><\/li>\n<\/ol>\n<p>now that we have found the derivative correctly, let&#8217;s simplify the final answer using algebra.<\/p>\n<p><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/chain_rule_example_solution.gif\" alt=\"final simplified form of the example derivative\" width=\"358\" height=\"43\" class=\"aligncenter size-full wp-image-9302\" \/><\/p>\n<h2>conclusion<\/h2>\n<p>use the chain rule to find the derivative of a composition.  always keep in mind the <em>3-step process<\/em>:<\/p>\n<ol>\n<li>find the derivative of the outside.<\/li>\n<li>plug in the inside function.<\/li>\n<li>multiply by the derivative of the inside.<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>of all the derivative rules it seems that the chain rule gets the worst press. many students dread the rule, think that it&#8217;s too difficult, don&#8217;t fully understand where to apply it, and generally wish that it would go away. are you in this group? if so then i hope that by the end of [&hellip;]<\/p>\n","protected":false},"author":223,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[240],"tags":[241],"ppma_author":[24932],"class_list":["post-9290","post","type-post","status-publish","format-standard","hentry","category-ap","tag-ap-calculus"],"acf":[],"yoast_head":"<!-- this site is optimized with the yoast seo premium plugin v21.7 (yoast seo v21.7) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>ap calculus review: chain rule - magoosh blog | high school<\/title>\n<meta name=\"description\" content=\"do you dread the chain rule, think that it&#039;s too difficult, or don&#039;t fully understand where to apply it? click here for a quick review of the chain rule.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-chain-rule\/\" \/>\n<meta property=\"og:locale\" content=\"en_us\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"ap calculus review: chain rule\" \/>\n<meta property=\"og:description\" content=\"do you dread the chain rule, think that it&#039;s too difficult, or don&#039;t fully understand where to apply it? click here for a quick review of the chain rule.\" \/>\n<meta property=\"og:url\" content=\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-chain-rule\/\" \/>\n<meta property=\"og:site_name\" content=\"magoosh blog | high school\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/magooshsat\/\" \/>\n<meta property=\"article:published_time\" content=\"2017-03-14t17:31:17+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2017-03-12t17:32:11+00:00\" \/>\n<meta property=\"og:image\" content=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/chainrule.gif\" \/>\n<meta name=\"author\" content=\"shaun ault\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@shaunaultmath\" \/>\n<meta name=\"twitter:site\" content=\"@magooshsat_act\" \/>\n<meta name=\"twitter:label1\" content=\"written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"shaun ault\" \/>\n\t<meta name=\"twitter:label2\" content=\"est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"article\",\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-chain-rule\/#article\",\"ispartof\":{\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-chain-rule\/\"},\"author\":{\"name\":\"shaun ault\",\"@id\":\"\/\/www.catharsisit.com\/hs\/#\/schema\/person\/f01e70874cef77d6f6392c12c43f6b6f\"},\"headline\":\"ap calculus review: chain rule\",\"datepublished\":\"2017-03-14t17:31:17+00:00\",\"datemodified\":\"2017-03-12t17:32:11+00:00\",\"mainentityofpage\":{\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-chain-rule\/\"},\"wordcount\":467,\"commentcount\":0,\"publisher\":{\"@id\":\"\/\/www.catharsisit.com\/hs\/#organization\"},\"keywords\":[\"ap calculus\"],\"articlesection\":[\"ap\"],\"inlanguage\":\"en-us\",\"potentialaction\":[{\"@type\":\"commentaction\",\"name\":\"comment\",\"target\":[\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-chain-rule\/#respond\"]}]},{\"@type\":\"webpage\",\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-chain-rule\/\",\"url\":\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-chain-rule\/\",\"name\":\"ap calculus review: chain rule - magoosh blog | high school\",\"ispartof\":{\"@id\":\"\/\/www.catharsisit.com\/hs\/#website\"},\"datepublished\":\"2017-03-14t17:31:17+00:00\",\"datemodified\":\"2017-03-12t17:32:11+00:00\",\"description\":\"do you dread the chain rule, think that it's too difficult, or don't fully understand where to apply it? click here for a quick review of the chain rule.\",\"breadcrumb\":{\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-chain-rule\/#breadcrumb\"},\"inlanguage\":\"en-us\",\"potentialaction\":[{\"@type\":\"readaction\",\"target\":[\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-chain-rule\/\"]}]},{\"@type\":\"breadcrumblist\",\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-chain-rule\/#breadcrumb\",\"itemlistelement\":[{\"@type\":\"listitem\",\"position\":1,\"name\":\"home\",\"item\":\"\/\/www.catharsisit.com\/hs\/\"},{\"@type\":\"listitem\",\"position\":2,\"name\":\"ap calculus review: chain rule\"}]},{\"@type\":\"website\",\"@id\":\"\/\/www.catharsisit.com\/hs\/#website\",\"url\":\"\/\/www.catharsisit.com\/hs\/\",\"name\":\"magoosh blog | high school\",\"description\":\"act, sat, college admissions, life\",\"publisher\":{\"@id\":\"\/\/www.catharsisit.com\/hs\/#organization\"},\"potentialaction\":[{\"@type\":\"searchaction\",\"target\":{\"@type\":\"entrypoint\",\"urltemplate\":\"\/\/www.catharsisit.com\/hs\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inlanguage\":\"en-us\"},{\"@type\":\"organization\",\"@id\":\"\/\/www.catharsisit.com\/hs\/#organization\",\"name\":\"magoosh\",\"url\":\"\/\/www.catharsisit.com\/hs\/\",\"logo\":{\"@type\":\"imageobject\",\"inlanguage\":\"en-us\",\"@id\":\"\/\/www.catharsisit.com\/hs\/#\/schema\/logo\/image\/\",\"url\":\"\/\/www.catharsisit.com\/hs\/files\/2019\/02\/magoosh-logo-purple-60h.png\",\"contenturl\":\"\/\/www.catharsisit.com\/hs\/files\/2019\/02\/magoosh-logo-purple-60h.png\",\"width\":265,\"height\":60,\"caption\":\"magoosh\"},\"image\":{\"@id\":\"\/\/www.catharsisit.com\/hs\/#\/schema\/logo\/image\/\"},\"sameas\":[\"https:\/\/www.facebook.com\/magooshsat\/\",\"https:\/\/twitter.com\/magooshsat_act\"]},{\"@type\":\"person\",\"@id\":\"\/\/www.catharsisit.com\/hs\/#\/schema\/person\/f01e70874cef77d6f6392c12c43f6b6f\",\"name\":\"shaun ault\",\"image\":{\"@type\":\"imageobject\",\"inlanguage\":\"en-us\",\"@id\":\"\/\/www.catharsisit.com\/hs\/#\/schema\/person\/image\/7c80c2046678e19beb5b9c8401d56613\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/28c36635b8fed6717752755f46cda239?s=96&d=mm&r=g\",\"contenturl\":\"https:\/\/secure.gravatar.com\/avatar\/28c36635b8fed6717752755f46cda239?s=96&d=mm&r=g\",\"caption\":\"shaun ault\"},\"description\":\"shaun earned his ph. d. in mathematics from the ohio state university in 2008 (go bucks!!). he received his ba in mathematics with a minor in computer science from oberlin college in 2002. in addition, shaun earned a b. mus. from the oberlin conservatory in the same year, with a major in music composition. shaun still loves music -- almost as much as math! -- and he (thinks he) can play piano, guitar, and bass. shaun has taught and tutored students in mathematics for about a decade, and hopes his experience can help you to succeed!\",\"sameas\":[\"http:\/\/valdosta.academia.edu\/shaunault\",\"https:\/\/twitter.com\/shaunaultmath\"],\"url\":\"\/\/www.catharsisit.com\/hs\/author\/shaunault\/\"}]}<\/script>\n<!-- \/ yoast seo premium plugin. -->","yoast_head_json":{"title":"ap calculus review: chain rule - magoosh blog | high school","description":"do you dread the chain rule, think that it's too difficult, or don't fully understand where to apply it? click here for a quick review of the chain rule.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-chain-rule\/","og_locale":"en_us","og_type":"article","og_title":"ap calculus review: chain rule","og_description":"do you dread the chain rule, think that it's too difficult, or don't fully understand where to apply it? click here for a quick review of the chain rule.","og_url":"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-chain-rule\/","og_site_name":"magoosh blog | high school","article_publisher":"https:\/\/www.facebook.com\/magooshsat\/","article_published_time":"2017-03-14t17:31:17+00:00","article_modified_time":"2017-03-12t17:32:11+00:00","og_image":[{"url":"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/chainrule.gif"}],"author":"shaun ault","twitter_card":"summary_large_image","twitter_creator":"@shaunaultmath","twitter_site":"@magooshsat_act","twitter_misc":{"written by":"shaun ault","est. reading time":"2 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"article","@id":"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-chain-rule\/#article","ispartof":{"@id":"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-chain-rule\/"},"author":{"name":"shaun ault","@id":"\/\/www.catharsisit.com\/hs\/#\/schema\/person\/f01e70874cef77d6f6392c12c43f6b6f"},"headline":"ap calculus review: chain rule","datepublished":"2017-03-14t17:31:17+00:00","datemodified":"2017-03-12t17:32:11+00:00","mainentityofpage":{"@id":"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-chain-rule\/"},"wordcount":467,"commentcount":0,"publisher":{"@id":"\/\/www.catharsisit.com\/hs\/#organization"},"keywords":["ap calculus"],"articlesection":["ap"],"inlanguage":"en-us","potentialaction":[{"@type":"commentaction","name":"comment","target":["\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-chain-rule\/#respond"]}]},{"@type":"webpage","@id":"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-chain-rule\/","url":"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-chain-rule\/","name":"ap calculus review: chain rule - magoosh blog | high school","ispartof":{"@id":"\/\/www.catharsisit.com\/hs\/#website"},"datepublished":"2017-03-14t17:31:17+00:00","datemodified":"2017-03-12t17:32:11+00:00","description":"do you dread the chain rule, think that it's too difficult, or don't fully understand where to apply it? click here for a quick review of the chain rule.","breadcrumb":{"@id":"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-chain-rule\/#breadcrumb"},"inlanguage":"en-us","potentialaction":[{"@type":"readaction","target":["\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-chain-rule\/"]}]},{"@type":"breadcrumblist","@id":"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-chain-rule\/#breadcrumb","itemlistelement":[{"@type":"listitem","position":1,"name":"home","item":"\/\/www.catharsisit.com\/hs\/"},{"@type":"listitem","position":2,"name":"ap calculus review: chain rule"}]},{"@type":"website","@id":"\/\/www.catharsisit.com\/hs\/#website","url":"\/\/www.catharsisit.com\/hs\/","name":"magoosh blog | high school","description":"act, sat, college admissions, life","publisher":{"@id":"\/\/www.catharsisit.com\/hs\/#organization"},"potentialaction":[{"@type":"searchaction","target":{"@type":"entrypoint","urltemplate":"\/\/www.catharsisit.com\/hs\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inlanguage":"en-us"},{"@type":"organization","@id":"\/\/www.catharsisit.com\/hs\/#organization","name":"magoosh","url":"\/\/www.catharsisit.com\/hs\/","logo":{"@type":"imageobject","inlanguage":"en-us","@id":"\/\/www.catharsisit.com\/hs\/#\/schema\/logo\/image\/","url":"\/\/www.catharsisit.com\/hs\/files\/2019\/02\/magoosh-logo-purple-60h.png","contenturl":"\/\/www.catharsisit.com\/hs\/files\/2019\/02\/magoosh-logo-purple-60h.png","width":265,"height":60,"caption":"magoosh"},"image":{"@id":"\/\/www.catharsisit.com\/hs\/#\/schema\/logo\/image\/"},"sameas":["https:\/\/www.facebook.com\/magooshsat\/","https:\/\/twitter.com\/magooshsat_act"]},{"@type":"person","@id":"\/\/www.catharsisit.com\/hs\/#\/schema\/person\/f01e70874cef77d6f6392c12c43f6b6f","name":"shaun ault","image":{"@type":"imageobject","inlanguage":"en-us","@id":"\/\/www.catharsisit.com\/hs\/#\/schema\/person\/image\/7c80c2046678e19beb5b9c8401d56613","url":"https:\/\/secure.gravatar.com\/avatar\/28c36635b8fed6717752755f46cda239?s=96&d=mm&r=g","contenturl":"https:\/\/secure.gravatar.com\/avatar\/28c36635b8fed6717752755f46cda239?s=96&d=mm&r=g","caption":"shaun ault"},"description":"shaun earned his ph. d. in mathematics from the ohio state university in 2008 (go bucks!!). he received his ba in mathematics with a minor in computer science from oberlin college in 2002. in addition, shaun earned a b. mus. from the oberlin conservatory in the same year, with a major in music composition. shaun still loves music -- almost as much as math! -- and he (thinks he) can play piano, guitar, and bass. shaun has taught and tutored students in mathematics for about a decade, and hopes his experience can help you to succeed!","sameas":["http:\/\/valdosta.academia.edu\/shaunault","https:\/\/twitter.com\/shaunaultmath"],"url":"\/\/www.catharsisit.com\/hs\/author\/shaunault\/"}]}},"authors":[{"term_id":24932,"user_id":223,"is_guest":0,"slug":"shaunault","display_name":"shaun ault","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/28c36635b8fed6717752755f46cda239?s=96&d=mm&r=g","user_url":"http:\/\/valdosta.academia.edu\/shaunault","last_name":"ault","first_name":"shaun","description":"shaun earned his ph. d. in mathematics from the ohio state university in 2008 (go bucks!!). he received his ba in mathematics with a minor in computer science from oberlin college in 2002. in addition, shaun earned a b. mus. from the oberlin conservatory in the same year, with a major in music composition.  shaun still loves music -- almost as much as math! -- and he (thinks he) can play piano, guitar, and bass.  shaun has taught and tutored students in mathematics for about a decade, and hopes his experience can help you to succeed!"}],"_links":{"self":[{"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/posts\/9290"}],"collection":[{"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/users\/223"}],"replies":[{"embeddable":true,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/comments?post=9290"}],"version-history":[{"count":0,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/posts\/9290\/revisions"}],"wp:attachment":[{"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/media?parent=9290"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/categories?post=9290"},{"taxonomy":"post_tag","embeddable":true,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/tags?post=9290"},{"taxonomy":"author","embeddable":true,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/ppma_author?post=9290"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}