{"id":240310,"date":"2025-07-03T08:28:32","date_gmt":"2025-07-03T08:28:32","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=240310"},"modified":"2025-07-03T08:28:34","modified_gmt":"2025-07-03T08:28:34","slug":"find-the-derivative-of-the-function","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/03\/find-the-derivative-of-the-function\/","title":{"rendered":"Find the derivative of the function"},"content":{"rendered":"\n

Find the derivative of the function. y = x sec(2x) \\frac{dy}{dx} =<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

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

To differentiate the function y=x\u22c5sec\u2061(2x)y = x \\cdot \\sec(2x)y=x\u22c5sec(2x), we need to apply both the product rule and the chain rule.<\/p>\n\n\n\n

Step 1: Apply the product rule<\/h3>\n\n\n\n

The product rule states that if y=u\u22c5vy = u \\cdot vy=u\u22c5v, then:dydx=u\u2032\u22c5v+u\u22c5v\u2032\\frac{dy}{dx} = u’ \\cdot v + u \\cdot v’dxdy\u200b=u\u2032\u22c5v+u\u22c5v\u2032<\/p>\n\n\n\n

Here, u=xu = xu=x and v=sec\u2061(2x)v = \\sec(2x)v=sec(2x). We need to find the derivatives of uuu and vvv.<\/p>\n\n\n\n

    \n
  • The derivative of u=xu = xu=x is u\u2032=1u’ = 1u\u2032=1.<\/li>\n\n\n\n
  • The derivative of v=sec\u2061(2x)v = \\sec(2x)v=sec(2x) requires the chain rule.<\/li>\n<\/ul>\n\n\n\n

    Step 2: Apply the chain rule to v=sec\u2061(2x)v = \\sec(2x)v=sec(2x)<\/h3>\n\n\n\n

    The chain rule states that the derivative of a composite function f(g(x))f(g(x))f(g(x)) is:ddx[f(g(x))]=f\u2032(g(x))\u22c5g\u2032(x)\\frac{d}{dx} [f(g(x))] = f'(g(x)) \\cdot g'(x)dxd\u200b[f(g(x))]=f\u2032(g(x))\u22c5g\u2032(x)<\/p>\n\n\n\n

    In this case, f(x)=sec\u2061(x)f(x) = \\sec(x)f(x)=sec(x), and the derivative of sec\u2061(x)\\sec(x)sec(x) is sec\u2061(x)tan\u2061(x)\\sec(x) \\tan(x)sec(x)tan(x). So, applying the chain rule:ddx[sec\u2061(2x)]=sec\u2061(2x)\u22c5tan\u2061(2x)\u22c5ddx[2x]=2\u22c5sec\u2061(2x)\u22c5tan\u2061(2x)\\frac{d}{dx} [\\sec(2x)] = \\sec(2x) \\cdot \\tan(2x) \\cdot \\frac{d}{dx}[2x] = 2 \\cdot \\sec(2x) \\cdot \\tan(2x)dxd\u200b[sec(2x)]=sec(2x)\u22c5tan(2x)\u22c5dxd\u200b[2x]=2\u22c5sec(2x)\u22c5tan(2x)<\/p>\n\n\n\n

    Step 3: Combine the results using the product rule<\/h3>\n\n\n\n

    Now, substitute everything back into the product rule:dydx=1\u22c5sec\u2061(2x)+x\u22c5(2\u22c5sec\u2061(2x)\u22c5tan\u2061(2x))\\frac{dy}{dx} = 1 \\cdot \\sec(2x) + x \\cdot \\left( 2 \\cdot \\sec(2x) \\cdot \\tan(2x) \\right)dxdy\u200b=1\u22c5sec(2x)+x\u22c5(2\u22c5sec(2x)\u22c5tan(2x))<\/p>\n\n\n\n

    Thus, the derivative is:dydx=sec\u2061(2x)+2x\u22c5sec\u2061(2x)\u22c5tan\u2061(2x)\\frac{dy}{dx} = \\sec(2x) + 2x \\cdot \\sec(2x) \\cdot \\tan(2x)dxdy\u200b=sec(2x)+2x\u22c5sec(2x)\u22c5tan(2x)<\/p>\n\n\n\n

    Conclusion:<\/h3>\n\n\n\n

    The derivative of the function y=x\u22c5sec\u2061(2x)y = x \\cdot \\sec(2x)y=x\u22c5sec(2x) is:dydx=sec\u2061(2x)+2x\u22c5sec\u2061(2x)\u22c5tan\u2061(2x)\\frac{dy}{dx} = \\sec(2x) + 2x \\cdot \\sec(2x) \\cdot \\tan(2x)dxdy\u200b=sec(2x)+2x\u22c5sec(2x)\u22c5tan(2x)<\/p>\n\n\n\n

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

    Find the derivative of the function. y = x sec(2x) \\frac{dy}{dx} = The Correct Answer and Explanation is: To differentiate the function y=x\u22c5sec\u2061(2x)y = x \\cdot \\sec(2x)y=x\u22c5sec(2x), we need to apply both the product rule and the chain rule. Step 1: Apply the product rule The product rule states that if y=u\u22c5vy = u \\cdot […]<\/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-240310","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/240310","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=240310"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/240310\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=240310"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=240310"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=240310"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}