{"id":241417,"date":"2025-07-03T17:46:36","date_gmt":"2025-07-03T17:46:36","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=241417"},"modified":"2025-07-03T17:46:38","modified_gmt":"2025-07-03T17:46:38","slug":"find-the-derivative-of-the-function-2","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/03\/find-the-derivative-of-the-function-2\/","title":{"rendered":"Find the derivative of the function"},"content":{"rendered":"\n

Find the derivative of the function: Simplify if possible – 3arctan(arctan)<\/p>\n\n\n\n

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

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

Of course. Here is the correct answer and a detailed explanation.<\/p>\n\n\n\n

Correct Answer:<\/strong>
The derivative of the function y = sqrt(3*arctan(x)) is:<\/p>\n\n\n\n

y’ = 3 \/ (2(1 + x^2) * sqrt(3*arctan(x)))<\/p>\n\n\n\n

Explanation:<\/h3>\n\n\n\n

To find the derivative of the function y = sqrt(3*arctan(x)), we need to use the Chain Rule. The Chain Rule is used for differentiating composite functions, which are functions nested inside other functions.<\/p>\n\n\n\n

1. Rewrite the Function<\/strong><\/p>\n\n\n\n

First, it is helpful to rewrite the square root as a fractional exponent. This makes applying the power rule more straightforward.<\/p>\n\n\n\n

y = (3*arctan(x))^(1\/2)<\/p>\n\n\n\n

2. Identify the Inner and Outer Functions<\/strong><\/p>\n\n\n\n

The function is a composition of two parts:<\/p>\n\n\n\n

    \n
  • The\u00a0outer function<\/strong>\u00a0is the square root part, which we can call\u00a0f(u) = u^(1\/2).<\/li>\n\n\n\n
  • The\u00a0inner function<\/strong>\u00a0is the part inside the square root, which we can call\u00a0g(x) = 3*arctan(x).<\/li>\n<\/ul>\n\n\n\n

    3. Apply the Chain Rule<\/strong><\/p>\n\n\n\n

    The Chain Rule states that the derivative of a composite function y = f(g(x)) is the derivative of the outer function with respect to the inner function, multiplied by the derivative of the inner function. The formula is:<\/p>\n\n\n\n

    y’ = f'(g(x)) * g'(x)<\/p>\n\n\n\n

    4. Find the Derivatives of the Parts<\/strong><\/p>\n\n\n\n

      \n
    • Derivative of the outer function:<\/strong>\u00a0We use the power rule to differentiate\u00a0f(u) = u^(1\/2).
      f'(u) = (1\/2) * u^(1\/2 – 1) = (1\/2) * u^(-1\/2) = 1 \/ (2 * u^(1\/2))<\/li>\n\n\n\n
    • Derivative of the inner function:<\/strong>\u00a0We differentiate\u00a0g(x) = 3*arctan(x). The derivative of\u00a0arctan(x)\u00a0is a standard result:\u00a01 \/ (1 + x^2).
      g'(x) = 3 * d\/dx(arctan(x)) = 3 * (1 \/ (1 + x^2)) = 3 \/ (1 + x^2)<\/li>\n<\/ul>\n\n\n\n

      5. Combine and Simplify<\/strong><\/p>\n\n\n\n

      Now, we substitute these pieces back into the Chain Rule formula:<\/p>\n\n\n\n

      y’ = [1 \/ (2 * (3*arctan(x))^(1\/2))] * [3 \/ (1 + x^2)]<\/p>\n\n\n\n

      Multiplying the numerators and denominators gives us the final simplified answer:<\/p>\n\n\n\n

      y’ = 3 \/ (2 * (1 + x^2) * sqrt(3*arctan(x)))<\/p>\n\n\n\n

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

      Find the derivative of the function: Simplify if possible – 3arctan(arctan) The Correct Answer and Explanation is: Of course. Here is the correct answer and a detailed explanation. Correct Answer:The derivative of the function y = sqrt(3*arctan(x)) is: y’ = 3 \/ (2(1 + x^2) * sqrt(3*arctan(x))) Explanation: To find the derivative of the function y = sqrt(3*arctan(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-241417","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/241417","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=241417"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/241417\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=241417"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=241417"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=241417"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}