{"id":243608,"date":"2025-07-04T15:19:23","date_gmt":"2025-07-04T15:19:23","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=243608"},"modified":"2025-07-04T15:19:28","modified_gmt":"2025-07-04T15:19:28","slug":"find-the-derivative-of-the-function-fx2ln%e2%81%a17x","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/04\/find-the-derivative-of-the-function-fx2ln%e2%81%a17x\/","title":{"rendered":"Find the derivative of the function\u00a0f(x)=2ln\u2061(7x)"},"content":{"rendered":"\n<h1 class=\"wp-block-heading\">Find the derivative of the function&nbsp;f(x)=2ln\u2061(7x)<\/h1>\n\n\n\n<p> <\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/image-139.png\" alt=\"\" class=\"wp-image-243677\"\/><\/figure>\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>The correct answer is f'(x) = 2\/x.<\/p>\n\n\n\n<p>To find the derivative of the function f(x) = 2 ln(7x), we need to apply the rules of differentiation, specifically the Constant Multiple Rule and the Chain Rule.<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Constant Multiple Rule<\/strong>: The function can be seen as a constant (2) multiplied by a function (ln(7x)). The Constant Multiple Rule states that the derivative of\u00a0c * g(x)\u00a0is\u00a0c * g'(x). Therefore, we can write:<br>f'(x) = 2 * d\/dx[ln(7x)]<\/li>\n\n\n\n<li><strong>Chain Rule<\/strong>: The term ln(7x) is a composite function. The outer function is the natural logarithm, ln(u), and the inner function is u = 7x. The Chain Rule states that the derivative of a composite function is the derivative of the outer function (with respect to the inner function) multiplied by the derivative of the inner function.<ul><li>The derivative of the outer function, ln(u), is 1\/u.<\/li><li>The derivative of the inner function, 7x, is 7.<\/li><\/ul>Applying the Chain Rule to ln(7x), we get:<br>d\/dx[ln(7x)] = (1 \/ 7x) * 7<\/li>\n\n\n\n<li><strong>Simplify and Combine<\/strong>: Now, we simplify the expression from the Chain Rule:<br>(1 \/ 7x) * 7 = 7 \/ (7x) = 1\/xFinally, we multiply this result by the constant 2 from the first step:<br>f'(x) = 2 * (1\/x) = 2\/x<\/li>\n<\/ol>\n\n\n\n<p><strong>Alternative Method (Using Logarithm Properties):<\/strong><\/p>\n\n\n\n<p>We can also simplify the function first using the logarithm property ln(ab) = ln(a) + ln(b):<br>f(x) = 2 ln(7x) = 2[ln(7) + ln(x)] = 2ln(7) + 2ln(x)<\/p>\n\n\n\n<p>Now, we can differentiate this expression term by term:<br>f'(x) = d\/dx[2ln(7)] + d\/dx[2ln(x)]<\/p>\n\n\n\n<p>Since 2ln(7) is a constant, its derivative is 0. The derivative of 2ln(x) is 2 * (1\/x) = 2\/x.<br>f'(x) = 0 + 2\/x = 2\/x<\/p>\n\n\n\n<p>Both methods yield the same result, confirming that the derivative of f(x) = 2 ln(7x) is f'(x) = 2\/x.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner5-348.jpeg\" alt=\"\" class=\"wp-image-243681\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Find the derivative of the function&nbsp;f(x)=2ln\u2061(7x) The Correct Answer and Explanation is: The correct answer is f'(x) = 2\/x. To find the derivative of the function f(x) = 2 ln(7x), we need to apply the rules of differentiation, specifically the Constant Multiple Rule and the Chain Rule. Alternative Method (Using Logarithm Properties): We can also [&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-243608","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/243608","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=243608"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/243608\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=243608"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=243608"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=243608"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}