{"id":154366,"date":"2022-03-25T23:27:19","date_gmt":"2022-03-25T17:57:19","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/leibnitz-theorem-derivation-solved-examples-and-faqs\/"},"modified":"2022-12-31T16:13:09","modified_gmt":"2022-12-31T10:43:09","slug":"leibnitz-theorem-derivation-solved-examples-and-faqs","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/maths\/leibnitz-theorem\/","title":{"rendered":"Leibnitz Theorem &#8211; Derivation, Solved Examples, and FAQs"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/leibnitz-theorem\/#Leibnitz_Theorem_For_nth_Derivative\" title=\"Leibnitz Theorem For nth Derivative\">Leibnitz Theorem For nth Derivative<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/leibnitz-theorem\/#Derivation_of_the_Leibnitz_TheoremFormula\" title=\"Derivation of the Leibnitz Theorem\/Formula\">Derivation of the Leibnitz Theorem\/Formula<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/leibnitz-theorem\/#Explain_in_Detail\" title=\"Explain in Detail :\">Explain in Detail :<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Leibnitz_Theorem_For_nth_Derivative\"><\/span>Leibnitz Theorem For nth Derivative<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>If \u0192 is a function and \u0192\u2019, \u0192\u201d, \u0192\u2019\u2019, \u0192\u2019\u2019\u2019, \u0192\u2019\u2019\u2019\u2019 are its derivatives at a point, then<\/p>\n<p>If \u0192 is a differentiable function, \u0192\u2019(x) exists at all points in the domain of \u0192, and \u0192 is continuous at a point c, then<\/p>\n<p>Theorem 1<\/p>\n<p>The nth derivative of \u0192 at c exists and is<\/p>\n<p>\u0192\u2019\u2019\u2019(c) = \u0192\u2019(c)<\/p>\n<p>Proof<\/p>\n<p>The proof is by induction on n. The base case is n = 1, for which<\/p>\n<p>\u0192\u2019\u2019\u2019(c) = \u0192\u2019(c)<\/p>\n<p>The inductive step is to show that<\/p>\n<p>\u0192\u2019\u2019\u2019(c) = \u0192\u2019\u2019(c) \u0192\u2019(c)<\/p>\n<p>which is the same as<\/p>\n<p>(\u0192\u2019\u2019\u2019)\u2019 = \u0192\u2019\u2019\u2019\u2019<\/p>\n<p>But<\/p>\n<p>(\u0192\u2019\u2019\u2019)\u2019 = \u0192\u2019\u2019\u2019<\/p>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-154365 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/leibnitz-theorem-derivation-solved-examples-and-faqs.jpg\" alt=\"\" width=\"606\" height=\"428\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/leibnitz-theorem-derivation-solved-examples-and-faqs.jpg?v=1648231034 606w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/leibnitz-theorem-derivation-solved-examples-and-faqs-300x212.jpg?v=1648231034 300w\" sizes=\"(max-width: 606px) 100vw, 606px\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Derivation_of_the_Leibnitz_TheoremFormula\"><\/span>Derivation of the Leibnitz Theorem\/Formula<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Let $f(x) = \\frac{x^2}{2}$. The derivative of $f$ is $f'(x) = 2x$.<\/p>\n<p>Now, let $g(x) = x^2$. The derivative of $g$ is $g'(x) = 2x$.<\/p>\n<p>We want to find the derivative of $f(g(x))$.<\/p>\n<p>This is equal to the derivative of $f$ multiplied by the derivative of $g$, or $f'(g(x)) = 2(g'(x))$.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Explain_in_Detail\"><\/span>Explain in Detail :<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The figure below shows a simple bar chart that compares the number of passengers carried by bus and train in a certain city over the past five years.<\/p>\n<p>The bar chart shows that the number of passengers carried by bus has been decreasing over the past five years, while the number of passengers carried by train has been increasing. In 2013, the number of passengers carried by bus was twice as high as the number of passengers carried by train, but by 2018, the number of passengers carried by train was twice as high as the number of passengers carried by bus.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Leibnitz Theorem For nth Derivative If \u0192 is a function and \u0192\u2019, \u0192\u201d, \u0192\u2019\u2019, \u0192\u2019\u2019\u2019, \u0192\u2019\u2019\u2019\u2019 are its derivatives at [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"Leibnitz Theorem","_yoast_wpseo_title":"","_yoast_wpseo_metadesc":"Learn about leibnitz theorem of maths in details explained by subject experts on infinitylearn.com. 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