{"id":231278,"date":"2025-06-10T18:43:49","date_gmt":"2025-06-10T18:43:49","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=231278"},"modified":"2025-06-10T18:43:51","modified_gmt":"2025-06-10T18:43:51","slug":"question-2-1-point-listen-find-the-derivative-of-the-function","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/10\/question-2-1-point-listen-find-the-derivative-of-the-function\/","title":{"rendered":"Question 2 (1 point) Listen Find the derivative of the function"},"content":{"rendered":"\n

Question 2 (1 point) Listen Find the derivative of the function.<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Answer<\/strong><\/h3>\n\n\n\n
f\u2032(x)=7(2x4\u22125x+1)6(8x3\u22125)f<\/em>\u2032(x<\/em>)=7(2x<\/em>4\u22125x<\/em>+1)6(8x<\/em>3\u22125)<\/code><\/pre>\n\n\n\n
Explanation<\/strong><\/h3>\n\n\n\n
The given function, <\/p>\n\n\n\n
f(x)=(2x4\u22125x+1)7f<\/em>(x<\/em>)=(2x<\/em>4\u22125x<\/em>+1)7<\/code><\/pre>\n\n\n\n
, is a composite function. To find its derivative, the Chain Rule must be applied. The Chain Rule is the fundamental method for differentiating a function nested inside another function. It states that the derivative of a composite function <\/p>\n\n\n\n
h(x)=g(u(x))h<\/em>(x<\/em>)=g<\/em>(u<\/em>(x<\/em>))<\/code><\/pre>\n\n\n\n
 is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. In formal notation, if <\/p>\n\n\n\n
f(x)=g(u(x))f<\/em>(x<\/em>)=g<\/em>(u<\/em>(x<\/em>))<\/code><\/pre>\n\n\n\n
, then <\/p>\n\n\n\n
f\u2032(x)=g\u2032(u(x))\u22c5u\u2032(x)f<\/em>\u2032(x<\/em>)=g<\/em>\u2032(u<\/em>(x<\/em>))\u22c5u<\/em>\u2032(x<\/em>)<\/code><\/pre>\n\n\n\n
.<\/p>\n\n\n\n
First, identify the outer and inner functions.
The outer function<\/strong> is the power of 7, which can be represented as <\/p>\n\n\n\n
g(u)=u7g<\/em>(u<\/em>)=u<\/em>7<\/code><\/pre>\n\n\n\n
.
The inner function<\/strong> is the polynomial inside the parentheses, <\/p>\n\n\n\n
u(x)=2x4\u22125x+1u<\/em>(x<\/em>)=2x<\/em>4\u22125x<\/em>+1<\/code><\/pre>\n\n\n\n
.<\/p>\n\n\n\n
Next, find the derivative of each of these functions separately.
The derivative of the outer function, <\/p>\n\n\n\n
g(u)=u7g<\/em>(u<\/em>)=u<\/em>7<\/code><\/pre>\n\n\n\n
, is found using the Power Rule. The derivative is <\/p>\n\n\n\n
g\u2032(u)=7u6g<\/em>\u2032(u<\/em>)=7u<\/em>6<\/code><\/pre>\n\n\n\n
.<\/p>\n\n\n\n
The derivative of the inner function, <\/p>\n\n\n\n
u(x)=2x4\u22125x+1u<\/em>(x<\/em>)=2x<\/em>4\u22125x<\/em>+1<\/code><\/pre>\n\n\n\n
, is found by applying the Power Rule to each term. The derivative is <\/p>\n\n\n\n
u\u2032(x)=2(4x3)\u22125(1)+0u<\/em>\u2032(x<\/em>)=2(4x<\/em>3)\u22125(1)+0<\/code><\/pre>\n\n\n\n
, which simplifies to <\/p>\n\n\n\n
u\u2032(x)=8x3\u22125u<\/em>\u2032(x<\/em>)=8x<\/em>3\u22125<\/code><\/pre>\n\n\n\n
.<\/p>\n\n\n\n
Finally, assemble these components according to the Chain Rule formula, <\/p>\n\n\n\n
f\u2032(x)=g\u2032(u(x))\u22c5u\u2032(x)f<\/em>\u2032(x<\/em>)=g<\/em>\u2032(u<\/em>(x<\/em>))\u22c5u<\/em>\u2032(x<\/em>)<\/code><\/pre>\n\n\n\n
. Substitute the expression for the inner function, <\/p>\n\n\n\n
u(x)u<\/em>(x<\/em>)<\/code><\/pre>\n\n\n\n
, back into the derivative of the outer function, <\/p>\n\n\n\n
g\u2032(u)g<\/em>\u2032(u<\/em>)<\/code><\/pre>\n\n\n\n
, and multiply by the derivative of the inner function, <\/p>\n\n\n\n
u\u2032(x)u<\/em>\u2032(x<\/em>)<\/code><\/pre>\n\n\n\n
.<\/p>\n\n\n\n
This yields:<\/p>\n\n\n\n
f\u2032(x)=7(2x4\u22125x+1)6\u22c5(8x3\u22125)f<\/em>\u2032(x<\/em>)=7(2x<\/em>4\u22125x<\/em>+1)6\u22c5(8x<\/em>3\u22125)<\/code><\/pre>\n\n\n\n
This expression is the final derivative of the original function. It is conventional to write the polynomial factor after the composite term.<\/p>\n\n\n\n
\"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"
Question 2 (1 point) Listen Find the derivative of the function. The Correct Answer and Explanation is: Answer Explanation The given function,  , is a composite function. To find its derivative, the Chain Rule must be applied. The Chain Rule is the fundamental method for differentiating a function nested inside another function. It states that […]<\/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-231278","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/231278","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=231278"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/231278\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=231278"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=231278"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=231278"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}