{"id":243708,"date":"2025-07-04T15:31:16","date_gmt":"2025-07-04T15:31:16","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=243708"},"modified":"2025-07-04T15:31:18","modified_gmt":"2025-07-04T15:31:18","slug":"find-the-20th-derivative-of-y-cos2x","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/04\/find-the-20th-derivative-of-y-cos2x\/","title":{"rendered":"Find the 20th derivative of Y cos(2x)."},"content":{"rendered":"\n

Find the 20th derivative of Y cos(2x). f (20)(x)<\/p>\n\n\n\n

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

To find the 20th derivative of the function y=cos\u2061(2x)y = \\cos(2x)y=cos(2x), we can first explore the pattern in the derivatives of cosine functions, especially those with arguments involving multiples of xxx.<\/p>\n\n\n\n

Step-by-Step Approach:<\/h3>\n\n\n\n

The function is:y=cos\u2061(2x)y = \\cos(2x)y=cos(2x)<\/p>\n\n\n\n

We begin by calculating the first few derivatives of y=cos\u2061(2x)y = \\cos(2x)y=cos(2x).<\/p>\n\n\n\n

    \n
  1. First derivative<\/strong>: ddx[cos\u2061(2x)]=\u22122sin\u2061(2x)\\frac{d}{dx} [\\cos(2x)] = -2\\sin(2x)dxd\u200b[cos(2x)]=\u22122sin(2x)<\/li>\n\n\n\n
  2. Second derivative<\/strong>: ddx[\u22122sin\u2061(2x)]=\u22122\u22c52cos\u2061(2x)=\u22124cos\u2061(2x)\\frac{d}{dx} [-2\\sin(2x)] = -2 \\cdot 2\\cos(2x) = -4\\cos(2x)dxd\u200b[\u22122sin(2x)]=\u22122\u22c52cos(2x)=\u22124cos(2x)<\/li>\n\n\n\n
  3. Third derivative<\/strong>: ddx[\u22124cos\u2061(2x)]=4\u22c52sin\u2061(2x)=8sin\u2061(2x)\\frac{d}{dx} [-4\\cos(2x)] = 4 \\cdot 2\\sin(2x) = 8\\sin(2x)dxd\u200b[\u22124cos(2x)]=4\u22c52sin(2x)=8sin(2x)<\/li>\n\n\n\n
  4. Fourth derivative<\/strong>: ddx[8sin\u2061(2x)]=8\u22c52cos\u2061(2x)=16cos\u2061(2x)\\frac{d}{dx} [8\\sin(2x)] = 8 \\cdot 2\\cos(2x) = 16\\cos(2x)dxd\u200b[8sin(2x)]=8\u22c52cos(2x)=16cos(2x)<\/li>\n<\/ol>\n\n\n\n

    Notice that after every 4th derivative, the function repeats in a similar form, with an increasing factor. So, the derivatives of y=cos\u2061(2x)y = \\cos(2x)y=cos(2x) follow a cyclical pattern with a period of 4. In general:<\/p>\n\n\n\n

      \n
    • The 0th derivative is cos\u2061(2x)\\cos(2x)cos(2x).<\/li>\n\n\n\n
    • The 1st derivative is \u22122sin\u2061(2x)-2\\sin(2x)\u22122sin(2x).<\/li>\n\n\n\n
    • The 2nd derivative is \u22124cos\u2061(2x)-4\\cos(2x)\u22124cos(2x).<\/li>\n\n\n\n
    • The 3rd derivative is 8sin\u2061(2x)8\\sin(2x)8sin(2x).<\/li>\n\n\n\n
    • The 4th derivative is 16cos\u2061(2x)16\\cos(2x)16cos(2x).<\/li>\n\n\n\n
    • This pattern repeats every 4 derivatives.<\/li>\n<\/ul>\n\n\n\n

      Applying this to the 20th derivative:<\/h3>\n\n\n\n

      Since the pattern repeats every 4 derivatives, we find the remainder when 20 is divided by 4:20\u00f74=5 remainder 020 \\div 4 = 5 \\text{ remainder } 020\u00f74=5 remainder 0<\/p>\n\n\n\n

      Thus, the 20th derivative corresponds to the 0th derivative in the cycle. The 0th derivative is simply the original function, cos\u2061(2x)\\cos(2x)cos(2x), multiplied by a constant factor, as seen in the pattern.<\/p>\n\n\n\n

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

      The 20th derivative of y=cos\u2061(2x)y = \\cos(2x)y=cos(2x) is:f(20)(x)=220cos\u2061(2x)f^{(20)}(x) = 2^{20} \\cos(2x)f(20)(x)=220cos(2x)<\/p>\n\n\n\n

      This result comes from the fact that the derivative increases by a factor of 2 with each step in the cycle, and after 20 derivatives (5 full cycles), the constant factor is 2202^{20}220.<\/p>\n\n\n\n

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

      Find the 20th derivative of Y cos(2x). f (20)(x) The Correct Answer and Explanation is: To find the 20th derivative of the function y=cos\u2061(2x)y = \\cos(2x)y=cos(2x), we can first explore the pattern in the derivatives of cosine functions, especially those with arguments involving multiples of xxx. Step-by-Step Approach: The function is:y=cos\u2061(2x)y = \\cos(2x)y=cos(2x) We begin […]<\/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-243708","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/243708","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=243708"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/243708\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=243708"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=243708"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=243708"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}