{"id":243687,"date":"2025-07-04T15:26:14","date_gmt":"2025-07-04T15:26:14","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=243687"},"modified":"2025-07-04T15:26:16","modified_gmt":"2025-07-04T15:26:16","slug":"find-the-derivative-of-the-function-3","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/04\/find-the-derivative-of-the-function-3\/","title":{"rendered":"Find the derivative of the function."},"content":{"rendered":"\n

Find the derivative of the function. f(x) = 7×2 ln(7x)<\/p>\n\n\n\n

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

To find the derivative of the function f(x)=7x2ln\u2061(7x)f(x) = 7x^2 \\ln(7x)f(x)=7x2ln(7x), we\u2019ll use the product rule<\/strong> since the function is the product of two functions: 7x27x^27×2 and ln\u2061(7x)\\ln(7x)ln(7x). The product rule states: ddx[u(x)v(x)]=u\u2032(x)v(x)+u(x)v\u2032(x)\\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)dxd\u200b[u(x)v(x)]=u\u2032(x)v(x)+u(x)v\u2032(x)<\/p>\n\n\n\n

Here, u(x)=7x2u(x) = 7x^2u(x)=7×2 and v(x)=ln\u2061(7x)v(x) = \\ln(7x)v(x)=ln(7x).<\/p>\n\n\n\n

Step 1: Differentiate u(x)=7x2u(x) = 7x^2u(x)=7×2<\/h3>\n\n\n\n

The derivative of 7x27x^27×2 is straightforward. Using the power rule: u\u2032(x)=14xu'(x) = 14xu\u2032(x)=14x<\/p>\n\n\n\n

Step 2: Differentiate v(x)=ln\u2061(7x)v(x) = \\ln(7x)v(x)=ln(7x)<\/h3>\n\n\n\n

To differentiate ln\u2061(7x)\\ln(7x)ln(7x), we need the chain rule. First, recognize that ln\u2061(7x)\\ln(7x)ln(7x) can be written as ln\u2061(7)+ln\u2061(x)\\ln(7) + \\ln(x)ln(7)+ln(x). Since ln\u2061(7)\\ln(7)ln(7) is a constant, its derivative is zero, and we only need to differentiate ln\u2061(x)\\ln(x)ln(x). The derivative of ln\u2061(x)\\ln(x)ln(x) is 1x\\frac{1}{x}x1\u200b, so: v\u2032(x)=17x\u00d77=1xv'(x) = \\frac{1}{7x} \\times 7 = \\frac{1}{x}v\u2032(x)=7×1\u200b\u00d77=x1\u200b<\/p>\n\n\n\n

Step 3: Apply the Product Rule<\/h3>\n\n\n\n

Now, applying the product rule: f\u2032(x)=u\u2032(x)v(x)+u(x)v\u2032(x)f'(x) = u'(x)v(x) + u(x)v'(x)f\u2032(x)=u\u2032(x)v(x)+u(x)v\u2032(x)<\/p>\n\n\n\n

Substitute the values of u(x)u(x)u(x), u\u2032(x)u'(x)u\u2032(x), v(x)v(x)v(x), and v\u2032(x)v'(x)v\u2032(x): f'(x) = (14x)(\\ln(7x)) + (7x^2)\\left(\\frac{1}{x}\\)<\/p>\n\n\n\n

Simplifying the second term: f\u2032(x)=14xln\u2061(7x)+7xf'(x) = 14x \\ln(7x) + 7xf\u2032(x)=14xln(7x)+7x<\/p>\n\n\n\n

Final Answer<\/h3>\n\n\n\n

The derivative of the function is: f\u2032(x)=14xln\u2061(7x)+7xf'(x) = 14x \\ln(7x) + 7xf\u2032(x)=14xln(7x)+7x<\/p>\n\n\n\n

Explanation:<\/h3>\n\n\n\n
    \n
  1. Product Rule<\/strong>: Since the given function is a product of two functions 7x27x^27×2 and ln\u2061(7x)\\ln(7x)ln(7x), we applied the product rule to differentiate it.<\/li>\n\n\n\n
  2. Chain Rule<\/strong>: When differentiating ln\u2061(7x)\\ln(7x)ln(7x), we used the chain rule to account for the inner function 7x7x7x.<\/li>\n\n\n\n
  3. Simplification<\/strong>: After differentiating, we simplified the second term by canceling out xxx in the numerator and denominator.<\/li>\n<\/ol>\n\n\n\n

    This method allows us to break down the problem step by step and apply the necessary rules of differentiation.<\/p>\n\n\n\n

    \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

    Find the derivative of the function. f(x) = 7×2 ln(7x) The Correct Answer and Explanation is: To find the derivative of the function f(x)=7x2ln\u2061(7x)f(x) = 7x^2 \\ln(7x)f(x)=7x2ln(7x), we\u2019ll use the product rule since the function is the product of two functions: 7x27x^27×2 and ln\u2061(7x)\\ln(7x)ln(7x). The product rule states: ddx[u(x)v(x)]=u\u2032(x)v(x)+u(x)v\u2032(x)\\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)dxd\u200b[u(x)v(x)]=u\u2032(x)v(x)+u(x)v\u2032(x) Here, u(x)=7x2u(x) […]<\/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-243687","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/243687","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=243687"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/243687\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=243687"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=243687"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=243687"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}