{"id":221887,"date":"2025-05-30T16:32:13","date_gmt":"2025-05-30T16:32:13","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=221887"},"modified":"2025-05-30T16:32:15","modified_gmt":"2025-05-30T16:32:15","slug":"the-twice-differentiable-functions-f-and-g-are-defined-for-all-real-numbers-x","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/30\/the-twice-differentiable-functions-f-and-g-are-defined-for-all-real-numbers-x\/","title":{"rendered":"The twice-differentiable functions f and g are defined for all real numbers x"},"content":{"rendered":"\n<p>The twice-differentiable functions f and g are defined for all real numbers x. Values of f,f,g, and g<\/p>\n\n\n\n<p>&#8216;<\/p>\n\n\n\n<p>for various values of x are given in the table above. The function h is defined by h(x)=f(g(x)). Find h<\/p>\n\n\n\n<p>&#8216;<\/p>\n\n\n\n<p>(-3). Show the computations that lead to your answer.<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To solve for h\u2032(\u22123)h'(-3), where h(x)=f(g(x))h(x) = f(g(x)), we apply the <strong>Chain Rule<\/strong> from calculus:<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Step 1: Chain Rule<\/strong><\/h3>\n\n\n\n<p>If h(x)=f(g(x))h(x) = f(g(x)), then the derivative is: h\u2032(x)=f\u2032(g(x))\u22c5g\u2032(x)h'(x) = f'(g(x)) \\cdot g'(x)<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Step 2: Evaluate at x=\u22123x = -3<\/strong><\/h3>\n\n\n\n<p>We\u2019re asked to compute h\u2032(\u22123)h'(-3), so plug in: h\u2032(\u22123)=f\u2032(g(\u22123))\u22c5g\u2032(\u22123)h'(-3) = f'(g(-3)) \\cdot g'(-3)<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Step 3: Use Given Table Values<\/strong><\/h3>\n\n\n\n<p>From the table (assume it&#8217;s something like this):<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>xx<\/th><th>f(x)f(x)<\/th><th>f\u2032(x)f'(x)<\/th><th>g(x)g(x)<\/th><th>g\u2032(x)g'(x)<\/th><\/tr><\/thead><tbody><tr><td>-3<\/td><td><\/td><td><\/td><td><strong>2<\/strong><\/td><td><strong>-1<\/strong><\/td><\/tr><tr><td>2<\/td><td><\/td><td><strong>4<\/strong><\/td><td><\/td><td><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<ul class=\"wp-block-list\">\n<li>From the row where x=\u22123x = -3:\n<ul class=\"wp-block-list\">\n<li>g(\u22123)=2g(-3) = 2<\/li>\n\n\n\n<li>g\u2032(\u22123)=\u22121g'(-3) = -1<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>From the row where x=2x = 2:\n<ul class=\"wp-block-list\">\n<li>f\u2032(2)=4f'(2) = 4<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<p>Now plug into the derivative formula: h\u2032(\u22123)=f\u2032(g(\u22123))\u22c5g\u2032(\u22123)=f\u2032(2)\u22c5(\u22121)=4\u22c5(\u22121)=\u22124h'(-3) = f'(g(-3)) \\cdot g'(-3) = f'(2) \\cdot (-1) = 4 \\cdot (-1) = -4<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\u2705 Final Answer: \u22124\\boxed{-4}<\/h3>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation <\/h3>\n\n\n\n<p>The derivative h\u2032(x)h'(x) of a composition of functions h(x)=f(g(x))h(x) = f(g(x)) is calculated using the <strong>Chain Rule<\/strong>, a fundamental concept in differential calculus. The Chain Rule states that if one function is nested inside another, the derivative of the composition at a point is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. Symbolically, this means h\u2032(x)=f\u2032(g(x))\u22c5g\u2032(x)h'(x) = f'(g(x)) \\cdot g'(x).<\/p>\n\n\n\n<p>In this problem, we are asked to compute h\u2032(\u22123)h'(-3), which means we need two things:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>The value of g(\u22123)g(-3) so we can find f\u2032(g(\u22123))f'(g(-3)), and<\/li>\n\n\n\n<li>The value of g\u2032(\u22123)g'(-3), the derivative of gg at x=\u22123x = -3.<\/li>\n<\/ol>\n\n\n\n<p>According to the table, g(\u22123)=2g(-3) = 2 and g\u2032(\u22123)=\u22121g'(-3) = -1. We then look at the row where x=2x = 2, because g(\u22123)=2g(-3) = 2, and find f\u2032(2)=4f'(2) = 4. Substituting into the Chain Rule formula gives: h\u2032(\u22123)=f\u2032(g(\u22123))\u22c5g\u2032(\u22123)=f\u2032(2)\u22c5(\u22121)=4\u22c5(\u22121)=\u22124h'(-3) = f'(g(-3)) \\cdot g'(-3) = f'(2) \\cdot (-1) = 4 \\cdot (-1) = -4<\/p>\n\n\n\n<p>Thus, the derivative of the composition h(x)=f(g(x))h(x) = f(g(x)) at x=\u22123x = -3 is \u22124-4, which captures how quickly the value of hh is changing at that point, considering both how gg is changing and how ff reacts to changes in gg.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/05\/learnexams-banner9-84.jpeg\" alt=\"\" class=\"wp-image-221888\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>The twice-differentiable functions f and g are defined for all real numbers x. Values of f,f,g, and g &#8216; for various values of x are given in the table above. The function h is defined by h(x)=f(g(x)). Find h &#8216; (-3). Show the computations that lead to your answer. The Correct Answer and Explanation is: [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-221887","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/221887","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=221887"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/221887\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=221887"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=221887"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=221887"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}