{"id":192696,"date":"2025-02-18T06:20:37","date_gmt":"2025-02-18T06:20:37","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=192696"},"modified":"2025-02-18T06:20:40","modified_gmt":"2025-02-18T06:20:40","slug":"differentiate-the-following-functions-with-respect-to-x-i-sin-4x-ii-cos-x3-iii-x3-tan-x","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/18\/differentiate-the-following-functions-with-respect-to-x-i-sin-4x-ii-cos-x3-iii-x3-tan-x\/","title":{"rendered":"Differentiate the following functions with respect to x: (i) sin 4x (ii) cos x3 (iii) x3 + tan x"},"content":{"rendered":"\n<p>Differentiate the following functions with respect to x: (i) sin 4x (ii) cos x3 (iii) x3 + tan x<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>Here are the derivatives of the functions with respect to ( x ):<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>For ( f(x) = \\sin(4x) ):<\/strong><\/li>\n<\/ol>\n\n\n\n<p>The derivative of ( \\sin(4x) ) is calculated using the chain rule, where the derivative of ( \\sin(u) ) with respect to ( u ) is ( \\cos(u) ), and the derivative of ( 4x ) with respect to ( x ) is 4. Hence, the derivative is:<\/p>\n\n\n\n<p>[<br>\\frac{d}{dx} \\sin(4x) = 4 \\cos(4x)<br>]<\/p>\n\n\n\n<ol start=\"2\" class=\"wp-block-list\">\n<li><strong>For ( f(x) = \\cos(x^3) ):<\/strong><\/li>\n<\/ol>\n\n\n\n<p>Again, applying the chain rule, the derivative of ( \\cos(u) ) with respect to ( u ) is ( -\\sin(u) ), and the derivative of ( x^3 ) with respect to ( x ) is ( 3x^2 ). Therefore, the derivative is:<\/p>\n\n\n\n<p>[<br>\\frac{d}{dx} \\cos(x^3) = -3x^2 \\sin(x^3)<br>]<\/p>\n\n\n\n<ol start=\"3\" class=\"wp-block-list\">\n<li><strong>For ( f(x) = x^3 + \\tan(x) ):<\/strong><\/li>\n<\/ol>\n\n\n\n<p>For this function, we differentiate term by term. The derivative of ( x^3 ) is ( 3x^2 ), and the derivative of ( \\tan(x) ) is ( \\sec^2(x) ). Hence, the derivative is:<\/p>\n\n\n\n<p>[<br>\\frac{d}{dx} \\left(x^3 + \\tan(x)\\right) = 3x^2 + \\sec^2(x)<br>]<\/p>\n\n\n\n<p>Thus, the results for the derivatives are:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( 4 \\cos(4x) )<\/li>\n\n\n\n<li>( -3x^2 \\sin(x^3) )<\/li>\n\n\n\n<li>( 3x^2 + \\sec^2(x) )<\/li>\n<\/ul>\n\n\n\n<p>Let me generate an image of these derivatives for visual clarity.<\/p>\n\n\n\n<p>Here is the image displaying the derivatives of the three functions. Let me know if you&#8217;d like further explanations or have any other questions!<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/02\/image-894.png\" alt=\"\" class=\"wp-image-192697\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Differentiate the following functions with respect to x: (i) sin 4x (ii) cos x3 (iii) x3 + tan x The Correct Answer and Explanation is : Here are the derivatives of the functions with respect to ( x ): The derivative of ( \\sin(4x) ) is calculated using the chain rule, where the derivative of [&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-192696","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/192696","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=192696"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/192696\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=192696"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=192696"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=192696"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}