{"id":26846,"date":"2019-02-23T07:51:26","date_gmt":"2019-02-23T07:51:26","guid":{"rendered":"https:\/\/www.testingdocs.com\/questions\/?p=26846"},"modified":"2025-05-17T07:18:12","modified_gmt":"2025-05-17T07:18:12","slug":"chain-rule-in-calculus","status":"publish","type":"post","link":"https:\/\/www.testingdocs.com\/questions\/chain-rule-in-calculus\/","title":{"rendered":"Chain Rule in Calculus"},"content":{"rendered":"<h1>Chain Rule in Calculus<\/h1>\n<p>The <strong>Chain Rule<\/strong> in calculus is a fundamental rule for finding the derivative of a composition of two functions. If you have two functions, <em>f(x)<\/em> and <em>g(x)<\/em>, and you want to find the derivative of their composition, <em>f(g(x))<\/em>, the chain rule states that:<\/p>\n<pre><code class=\"\" data-line=\"\">f&#039;(g(x)) = f&#039;(g(x)) \u00d7 g&#039;(x)<\/code><\/pre>\n<h2>Explanation<\/h2>\n<p>&#8211; <em>f(g(x))<\/em> is the composition of the functions <em>f<\/em> and <em>g<\/em>.<\/p>\n<p>&#8211; The derivative of the composition is found by taking the derivative of the outer function <em>f<\/em> evaluated at <em>g(x)<\/em>, and multiplying it by the derivative of the inner function <em>g(x)<\/em>.<\/p>\n<h2>Example<\/h2>\n<p>Suppose we want to differentiate the function <em>h(x) = sin(x\u00b2)<\/em>.<\/p>\n<ol>\n<li>Identify the inner and outer functions:\n<ul>\n<li>The outer function is <em>f(u) = sin(u)<\/em>, where <em>u = x\u00b2<\/em>.<\/li>\n<li>The inner function is <em>g(x) = x\u00b2<\/em>.<\/li>\n<\/ul>\n<\/li>\n<li>Apply the chain rule:\n<pre><code class=\"\" data-line=\"\">f&#039;(x) = cos(x\u00b2) \u00d7 (d\/dx)(x\u00b2)<\/code><\/pre>\n<\/li>\n<li>The derivative of <em>x\u00b2<\/em> is <em>2x<\/em>, so:\n<pre><code class=\"\" data-line=\"\">f&#039;(x) = cos(x\u00b2) \u00d7 2x<\/code><\/pre>\n<\/li>\n<\/ol>\n<p>Thus, the derivative of <em>sin(x\u00b2)<\/em> is <em>2x cos(x\u00b2)<\/em>.<\/p>\n<p>The chain rule is essential for differentiating complex functions where one function is nested inside another. It helps us calculate derivatives of composite functions in an easy and structured way.<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Chain Rule in Calculus The Chain Rule in calculus is a fundamental rule for finding the derivative of a composition of two functions. If you have two functions, f(x) and g(x), and you want to find the derivative of their composition, f(g(x)), the chain rule states that: f&#039;(g(x)) = f&#039;(g(x)) \u00d7 g&#039;(x) Explanation &#8211; f(g(x)) [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[849],"tags":[],"class_list":["post-26846","post","type-post","status-publish","format-standard","hentry","category-math-questions","has-post-title","has-post-date","has-post-category","has-post-tag","has-post-comment","has-post-author",""],"_links":{"self":[{"href":"https:\/\/www.testingdocs.com\/questions\/wp-json\/wp\/v2\/posts\/26846","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.testingdocs.com\/questions\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.testingdocs.com\/questions\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.testingdocs.com\/questions\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.testingdocs.com\/questions\/wp-json\/wp\/v2\/comments?post=26846"}],"version-history":[{"count":3,"href":"https:\/\/www.testingdocs.com\/questions\/wp-json\/wp\/v2\/posts\/26846\/revisions"}],"predecessor-version":[{"id":27418,"href":"https:\/\/www.testingdocs.com\/questions\/wp-json\/wp\/v2\/posts\/26846\/revisions\/27418"}],"wp:attachment":[{"href":"https:\/\/www.testingdocs.com\/questions\/wp-json\/wp\/v2\/media?parent=26846"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.testingdocs.com\/questions\/wp-json\/wp\/v2\/categories?post=26846"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.testingdocs.com\/questions\/wp-json\/wp\/v2\/tags?post=26846"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}