{"id":8939,"date":"2017-02-10t19:39:54","date_gmt":"2017-02-11t03:39:54","guid":{"rendered":"\/\/www.catharsisit.com\/hs\/?p=8939"},"modified":"2022-06-14t19:39:54","modified_gmt":"2022-06-15t02:39:54","slug":"mastering-chain-rule","status":"publish","type":"post","link":"\/\/www.catharsisit.com\/hs\/ap\/mastering-chain-rule\/","title":{"rendered":"mastering the chain rule"},"content":{"rendered":"<p>for the ap calculus exam, whether it\u2019s calculus ab or calculus bc, being completely fluent in taking the derivative of almost any function is imperative.\u00a0 we should know how to use all the most common methods for taking the derive well enough that we don\u2019t have to think about it on the exam itself. \u00a0the last thing we want to get caught up in on the exam is forgetting one of our derivative taking methods, such as the chain rule.<\/p>\n<p>at minimum, you should know how to use:<\/p>\n<ul>\n<li>the power rule<\/li>\n<li>the chain rule<\/li>\n<li>the product rule<\/li>\n<li>the quotient rule<\/li>\n<\/ul>\n<p>the chain rule is one of the most important rules in this list because many function cans be thought of as functions of functions (of functions of functions).<\/p>\n<h2>what is the chain rule and when do we use it?<\/h2>\n<p><strong>\u00a0<\/strong><\/p>\n<p>the chain rule is used when we have a function which is of the form f(x) = g(h(x)).<\/p>\n<p>for example:<\/p>\n<p><img decoding=\"async\" class=\"size-full wp-image-8940 alignnone\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/screen-shot-2017-02-06-at-2.00.16-pm.png\" alt=\"chain rule\" width=\"87\" height=\"27\" \/><\/p>\n<p>our function <em>f(x)<\/em> is made up of two smaller functions.\u00a0 you can think of f(x) = g(h(x)), where h(x) = x<sup>2<\/sup> and g(x) = sin(x).\u00a0 we can find the derivative of each of these functions individually.\u00a0 the chain rule allows us to take the derivative of the entire thing.<\/p>\n<h2>the chain rule<\/h2>\n<p><strong>\u00a0<img decoding=\"async\" class=\"alignnone size-full wp-image-8941\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/screen-shot-2017-02-06-at-2.00.20-pm.png\" alt=\"an algebraic formula\" width=\"140\" height=\"43\" \/><\/strong><\/p>\n<p>in our above example, f(x) = sin(x<sup>2<\/sup>).\u00a0\u00a0 h(x) = x<sup>2<\/sup> , h\u2019(x) = 2x, g(x) = sin(x), and g\u2019(x) = cos(x).\u00a0 the derivative of h(x) can be solved with the power rule, and the derivative of g(x) is a common derivative.\u00a0 you can find a list of common derivatives as well as explanations of the other derivative rules in our <a href=\"\/\/www.catharsisit.com\/hs\/ap\/calculus-review-derivative-rules\/\">review of derivative rules<\/a>.<\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-8971\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/screen-shot-2017-02-06-at-2.06.03-pm.png\" alt=\"an algebraic function formula\" width=\"143\" height=\"48\" \/><\/p>\n<h3>the difficulty in using the chain rule:<\/h3>\n<p>implementing the chain rule is usually not difficult.\u00a0 the problem that many students have trouble with is trying to figure out which parts of the function are within other functions (i.e., in the above example, which part if g(x) and which part is h(x).<br \/>\nthis is going to be a rule which is very important to practice with over and over again.<\/p>\n<h2>a handful of example problems<\/h2>\n<p>here is a list of example problems of varying difficulties.\u00a0 you do not need a calculator to answer any of these questions.\u00a0 you\u2019ll find the solutions below.<\/p>\n<p><img decoding=\"async\" class=\"size-full wp-image-9070 alignnone\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/screen-shot-2017-02-10-at-10.26.08-pm.png\" alt=\"chain rule practice problems\" width=\"242\" height=\"537\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/screen-shot-2017-02-10-at-10.26.08-pm.png 242w, \/\/www.catharsisit.com\/hs\/files\/2017\/02\/screen-shot-2017-02-10-at-10.26.08-pm-135x300.png 135w\" sizes=\"(max-width: 242px) 100vw, 242px\" \/><\/p>\n<h2>solutions and explanations to example problems<\/h2>\n<p>&nbsp;<\/p>\n<p><strong>\u00a0<\/strong>1.<strong><img decoding=\"async\" class=\"alignnone size-full wp-image-8967\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/screen-shot-2017-02-06-at-2.02.05-pm.png\" alt=\"an algebraic function problem equation\" width=\"113\" height=\"28\" \/><\/strong><\/p>\n<p><strong>solution:<\/strong><\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-8968\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/screen-shot-2017-02-06-at-2.02.10-pm.png\" alt=\"an algebraic function solution\" width=\"174\" height=\"32\" \/><br \/>\n&nbsp;<\/p>\n<p>2.<img decoding=\"async\" class=\"alignnone size-full wp-image-8969\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/screen-shot-2017-02-06-at-2.02.15-pm.png\" alt=\"an algebraic function problem equation\" width=\"118\" height=\"45\" \/><\/p>\n<p><strong>solution:<\/strong><\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-8970\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/screen-shot-2017-02-06-at-2.02.19-pm.png\" alt=\"an algebraic function solution\" width=\"175\" height=\"34\" \/><br \/>\n&nbsp;<\/p>\n<p>3. <img decoding=\"async\" class=\"alignnone size-full wp-image-8952\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/screen-shot-2017-02-06-at-2.01.02-pm.png\" alt=\"an algebraic function problem equation\" width=\"142\" height=\"29\" \/><\/p>\n<p><strong>solution:<\/strong><\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-8951\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/screen-shot-2017-02-06-at-2.00.58-pm.png\" alt=\"an algebraic function solution\" width=\"242\" height=\"40\" \/><br \/>\n&nbsp;<\/p>\n<p>4. <img decoding=\"async\" class=\"alignnone size-full wp-image-8953\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/screen-shot-2017-02-06-at-2.01.08-pm.png\" alt=\"an algebraic function problem equation\" width=\"131\" height=\"32\" \/><\/p>\n<p>this one is thrown in purposely, even though it is not a chain rule problem.\u00a0 many students get confused between when to use the chain rule (when you have a function of a function), and when to use the product rule (when you have a function multiplied by a function).<\/p>\n<p><strong>solution:<\/strong><\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-8954\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/screen-shot-2017-02-06-at-2.01.11-pm.png\" alt=\"an algebraic function solution\" width=\"240\" height=\"42\" \/><\/p>\n<p><em>(this can be reduced to cos(2x) if you know your trig identities.)<\/em><\/p>\n<p>&nbsp;<\/p>\n<p>5. <img decoding=\"async\" class=\"alignnone size-full wp-image-8955\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/screen-shot-2017-02-06-at-2.01.17-pm.png\" alt=\"an algebraic function problem equation\" width=\"142\" height=\"33\" \/><\/p>\n<p>we can also use the chain rule in combination with the other rules.<\/p>\n<p><strong>solution:<\/strong><\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-8956\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/screen-shot-2017-02-06-at-2.01.20-pm.png\" alt=\"an algebraic function solution\" width=\"239\" height=\"35\" \/><br \/>\n&nbsp;<\/p>\n<p>6. <img decoding=\"async\" class=\"alignnone size-full wp-image-8957\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/screen-shot-2017-02-06-at-2.01.23-pm.png\" alt=\"an algebraic function problem equation\" width=\"120\" height=\"28\" \/><\/p>\n<p><strong>solution:<\/strong><\/p>\n<p><strong><img decoding=\"async\" class=\"alignnone size-full wp-image-8958\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/screen-shot-2017-02-06-at-2.01.27-pm.png\" alt=\"an algebraic function solution\" width=\"210\" height=\"31\" \/><\/strong><br \/>\n&nbsp;<\/p>\n<p>7. <img decoding=\"async\" class=\"alignnone size-full wp-image-8972\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/screen-shot-2017-02-06-at-2.11.06-pm.png\" alt=\"an algebraic function problem equation\" width=\"137\" height=\"30\" \/><\/p>\n<p>again, we can use chain rule in combination with other rules.<\/p>\n<p><strong>solution:<\/strong><\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-8962\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/screen-shot-2017-02-06-at-2.01.45-pm.png\" alt=\"an algebraic function solution\" width=\"296\" height=\"29\" \/><br \/>\n&nbsp;<\/p>\n<p>8. <img decoding=\"async\" class=\"alignnone size-full wp-image-8961\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/screen-shot-2017-02-06-at-2.01.40-pm.png\" alt=\"an algebraic function problem equation\" width=\"140\" height=\"22\" \/><\/p>\n<p>we can use the chain rule multiple times in a row as well.<\/p>\n<p><strong>solution:<\/strong><\/p>\n<p><strong><img decoding=\"async\" class=\"alignnone size-full wp-image-8962\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/screen-shot-2017-02-06-at-2.01.45-pm.png\" alt=\"an algebraic function solution\" width=\"296\" height=\"29\" \/><\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>9.\u00a0<img decoding=\"async\" class=\"alignnone size-full wp-image-8963\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/screen-shot-2017-02-06-at-2.01.52-pm.png\" alt=\"an algebraic function problem equation\" width=\"121\" height=\"25\" \/><\/p>\n<p><strong>solution:<\/strong><\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-8964\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/screen-shot-2017-02-06-at-2.01.56-pm.png\" alt=\"an algebraic function solution\" width=\"158\" height=\"28\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/screen-shot-2017-02-06-at-2.01.56-pm.png 158w, \/\/www.catharsisit.com\/hs\/files\/2017\/02\/screen-shot-2017-02-06-at-2.01.56-pm-150x28.png 150w\" sizes=\"(max-width: 158px) 100vw, 158px\" \/><br \/>\n&nbsp;<\/p>\n<p>10.\u00a0<img decoding=\"async\" class=\"alignnone size-full wp-image-8965\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/screen-shot-2017-02-06-at-2.02.00-pm.png\" alt=\"an algebraic function problem equation\" width=\"98\" height=\"30\" \/><\/p>\n<p><strong>solution:<\/strong><\/p>\n<p><em>f'(x) =\u00a0<\/em>5(x + 2)<sup>4<\/sup><\/p>\n<p>&nbsp;<\/p>\n<p>11. <em>f(x)<\/em> = (x + 2)<sup>2<\/sup><\/p>\n<p><strong>solution:<\/strong><\/p>\n<p><em>f'(x) = <\/em>2(x + 2)<\/p>\n<p>&nbsp;<\/p>\n<p>12. <img decoding=\"async\" class=\"alignnone size-full wp-image-8949\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/screen-shot-2017-02-06-at-2.00.53-pm.png\" alt=\"an algebraic function problem equation\" width=\"131\" height=\"37\" \/><\/p>\n<p>you will find that chain rule problems often involve sin, cos, and tangent, as we often use trigonometric along with other functions.<\/p>\n<p><strong>solution: <\/strong><\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-8950\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/screen-shot-2017-02-06-at-2.00.56-pm.png\" alt=\"an algebraic function solution equation\" width=\"160\" height=\"39\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>mastering the chain rule is incredibly important for success on the ap calculus exam.\u00a0 the key to studying the chain rule, as well as any of the differentiation rules, is to practice with it as much as possible.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>in this article, learn how to master the chain rule by learning how it works, with examples and solutions to chain rule derivative problems.<\/p>\n","protected":false},"author":224,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[240],"tags":[241],"ppma_author":[24930],"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>mastering the chain rule - magoosh blog | high school<\/title>\n<meta name=\"description\" content=\"in this article, learn how to master the chain rule by learning how it works, with examples and solutions to chain rule derivative problems.\" \/>\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\/mastering-chain-rule\/\" \/>\n<meta property=\"og:locale\" content=\"en_us\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"mastering the chain rule\" \/>\n<meta property=\"og:description\" content=\"in this article, learn how to master the chain rule by learning how it works, with examples and solutions to chain rule derivative problems.\" \/>\n<meta property=\"og:url\" content=\"\/\/www.catharsisit.com\/hs\/ap\/mastering-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=\"2017-02-11t03:39:54+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2022-06-15t02:39:54+00:00\" \/>\n<meta property=\"og:image\" content=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/screen-shot-2017-02-06-at-2.00.16-pm.png\" \/>\n<meta name=\"author\" content=\"zachary\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@magooshsat_act\" \/>\n<meta name=\"twitter:site\" content=\"@magooshsat_act\" \/>\n<meta name=\"twitter:label1\" content=\"written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"zachary\" \/>\n\t<meta name=\"twitter:label2\" content=\"est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"article\",\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/mastering-chain-rule\/#article\",\"ispartof\":{\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/mastering-chain-rule\/\"},\"author\":{\"name\":\"zachary\",\"@id\":\"\/\/www.catharsisit.com\/hs\/#\/schema\/person\/2bb580c8b05f06690589778ec3805636\"},\"headline\":\"mastering the chain rule\",\"datepublished\":\"2017-02-11t03:39:54+00:00\",\"datemodified\":\"2022-06-15t02:39:54+00:00\",\"mainentityofpage\":{\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/mastering-chain-rule\/\"},\"wordcount\":596,\"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\/mastering-chain-rule\/#respond\"]}]},{\"@type\":\"webpage\",\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/mastering-chain-rule\/\",\"url\":\"\/\/www.catharsisit.com\/hs\/ap\/mastering-chain-rule\/\",\"name\":\"mastering the chain rule - magoosh blog | high school\",\"ispartof\":{\"@id\":\"\/\/www.catharsisit.com\/hs\/#website\"},\"datepublished\":\"2017-02-11t03:39:54+00:00\",\"datemodified\":\"2022-06-15t02:39:54+00:00\",\"description\":\"in this article, learn how to master the chain rule by learning how it works, with examples and solutions to chain rule derivative problems.\",\"breadcrumb\":{\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/mastering-chain-rule\/#breadcrumb\"},\"inlanguage\":\"en-us\",\"potentialaction\":[{\"@type\":\"readaction\",\"target\":[\"\/\/www.catharsisit.com\/hs\/ap\/mastering-chain-rule\/\"]}]},{\"@type\":\"breadcrumblist\",\"@id\":\"\/\/www.catharsisit.com\/hs\/ap\/mastering-chain-rule\/#breadcrumb\",\"itemlistelement\":[{\"@type\":\"listitem\",\"position\":1,\"name\":\"home\",\"item\":\"\/\/www.catharsisit.com\/hs\/\"},{\"@type\":\"listitem\",\"position\":2,\"name\":\"mastering the 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\/2bb580c8b05f06690589778ec3805636\",\"name\":\"zachary\",\"image\":{\"@type\":\"imageobject\",\"inlanguage\":\"en-us\",\"@id\":\"\/\/www.catharsisit.com\/hs\/#\/schema\/person\/image\/7095938d1c7af05e974b245c5dda6b38\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/1f537b483a302f935510556c8311590a?s=96&d=mm&r=g\",\"contenturl\":\"https:\/\/secure.gravatar.com\/avatar\/1f537b483a302f935510556c8311590a?s=96&d=mm&r=g\",\"caption\":\"zachary\"},\"description\":\"zachary is a former mechanical engineer and current high school physics, math, and computer science teacher. he graduated from mcgill university in 2011 and spent time in the automotive industry in detroit before moving into education. he has been teaching and tutoring for the past five years, but you can also find him adventuring, reading, rock climbing, and traveling whenever the opportunity arises.\",\"url\":\"\/\/www.catharsisit.com\/hs\/author\/zacharyswarth\/\"}]}<\/script>\n<!-- \/ yoast seo premium plugin. -->","yoast_head_json":{"title":"mastering the chain rule - magoosh blog | high school","description":"in this article, learn how to master the chain rule by learning how it works, with examples and solutions to chain rule derivative problems.","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\/mastering-chain-rule\/","og_locale":"en_us","og_type":"article","og_title":"mastering the chain rule","og_description":"in this article, learn how to master the chain rule by learning how it works, with examples and solutions to chain rule derivative problems.","og_url":"\/\/www.catharsisit.com\/hs\/ap\/mastering-chain-rule\/","og_site_name":"magoosh blog | high school","article_publisher":"https:\/\/www.facebook.com\/magooshsat\/","article_published_time":"2017-02-11t03:39:54+00:00","article_modified_time":"2022-06-15t02:39:54+00:00","og_image":[{"url":"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/screen-shot-2017-02-06-at-2.00.16-pm.png"}],"author":"zachary","twitter_card":"summary_large_image","twitter_creator":"@magooshsat_act","twitter_site":"@magooshsat_act","twitter_misc":{"written by":"zachary","est. reading time":"3 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"article","@id":"\/\/www.catharsisit.com\/hs\/ap\/mastering-chain-rule\/#article","ispartof":{"@id":"\/\/www.catharsisit.com\/hs\/ap\/mastering-chain-rule\/"},"author":{"name":"zachary","@id":"\/\/www.catharsisit.com\/hs\/#\/schema\/person\/2bb580c8b05f06690589778ec3805636"},"headline":"mastering the chain rule","datepublished":"2017-02-11t03:39:54+00:00","datemodified":"2022-06-15t02:39:54+00:00","mainentityofpage":{"@id":"\/\/www.catharsisit.com\/hs\/ap\/mastering-chain-rule\/"},"wordcount":596,"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\/mastering-chain-rule\/#respond"]}]},{"@type":"webpage","@id":"\/\/www.catharsisit.com\/hs\/ap\/mastering-chain-rule\/","url":"\/\/www.catharsisit.com\/hs\/ap\/mastering-chain-rule\/","name":"mastering the chain rule - magoosh blog | high school","ispartof":{"@id":"\/\/www.catharsisit.com\/hs\/#website"},"datepublished":"2017-02-11t03:39:54+00:00","datemodified":"2022-06-15t02:39:54+00:00","description":"in this article, learn how to master the chain rule by learning how it works, with examples and solutions to chain rule derivative problems.","breadcrumb":{"@id":"\/\/www.catharsisit.com\/hs\/ap\/mastering-chain-rule\/#breadcrumb"},"inlanguage":"en-us","potentialaction":[{"@type":"readaction","target":["\/\/www.catharsisit.com\/hs\/ap\/mastering-chain-rule\/"]}]},{"@type":"breadcrumblist","@id":"\/\/www.catharsisit.com\/hs\/ap\/mastering-chain-rule\/#breadcrumb","itemlistelement":[{"@type":"listitem","position":1,"name":"home","item":"\/\/www.catharsisit.com\/hs\/"},{"@type":"listitem","position":2,"name":"mastering the 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\/2bb580c8b05f06690589778ec3805636","name":"zachary","image":{"@type":"imageobject","inlanguage":"en-us","@id":"\/\/www.catharsisit.com\/hs\/#\/schema\/person\/image\/7095938d1c7af05e974b245c5dda6b38","url":"https:\/\/secure.gravatar.com\/avatar\/1f537b483a302f935510556c8311590a?s=96&d=mm&r=g","contenturl":"https:\/\/secure.gravatar.com\/avatar\/1f537b483a302f935510556c8311590a?s=96&d=mm&r=g","caption":"zachary"},"description":"zachary is a former mechanical engineer and current high school physics, math, and computer science teacher. he graduated from mcgill university in 2011 and spent time in the automotive industry in detroit before moving into education. he has been teaching and tutoring for the past five years, but you can also find him adventuring, reading, rock climbing, and traveling whenever the opportunity arises.","url":"\/\/www.catharsisit.com\/hs\/author\/zacharyswarth\/"}]}},"authors":[{"term_id":24930,"user_id":224,"is_guest":0,"slug":"zacharyswarth","display_name":"zachary","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/1f537b483a302f935510556c8311590a?s=96&d=mm&r=g","user_url":"","last_name":"swarth","first_name":"zachary","description":"zachary is a former mechanical engineer and current high school physics, math, and computer science teacher.  he graduated from mcgill university in 2011 and spent time in the automotive industry in detroit before moving into education.  he has been teaching and tutoring for the past five years, but you can also find him adventuring, reading, rock climbing, and traveling whenever the opportunity arises."}],"_links":{"self":[{"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/posts\/8939"}],"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\/224"}],"replies":[{"embeddable":true,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/comments?post=8939"}],"version-history":[{"count":0,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/posts\/8939\/revisions"}],"wp:attachment":[{"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/media?parent=8939"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/categories?post=8939"},{"taxonomy":"post_tag","embeddable":true,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/tags?post=8939"},{"taxonomy":"author","embeddable":true,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/ppma_author?post=8939"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}