{"id":263856,"date":"2025-07-21T09:34:06","date_gmt":"2025-07-21T09:34:06","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=263856"},"modified":"2025-07-21T09:34:09","modified_gmt":"2025-07-21T09:34:09","slug":"calculate-the-derivative-10x-in-tdt-dx-29-using-part-2-of-the-fundamental-theorem-of-calculus_-enclose","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/21\/calculate-the-derivative-10x-in-tdt-dx-29-using-part-2-of-the-fundamental-theorem-of-calculus_-enclose\/","title":{"rendered":"‘Calculate the derivative 10x In (t)dt dx 29 using Part 2 of the Fundamental Theorem of Calculus_ Enclose"},"content":{"rendered":"\n

‘Calculate the derivative 10x In (t)dt dx 29 using Part 2 of the Fundamental Theorem of Calculus_ Enclose arguments of functions in parentheses_ For example; sin (2x) 10x In (t)dt = dx 29’<\/p>\n\n\n\n

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

We are asked to calculate the derivative of the integral\u222b10x29ln\u2061(t)\u2009dtwith respect to\u2009x.\\int_{10x}^{29} \\ln(t) \\, dt \\quad \\text{with respect to} \\, x.\u222b10×29\u200bln(t)dtwith respect tox.<\/p>\n\n\n\n

This is a problem that can be approached using Part 2 of the Fundamental Theorem of Calculus<\/strong>. Part 2 of this theorem states that if F(x)=\u222ba(x)b(x)f(t)\u2009dtF(x) = \\int_{a(x)}^{b(x)} f(t) \\, dtF(x)=\u222ba(x)b(x)\u200bf(t)dt, then the derivative of F(x)F(x)F(x) with respect to xxx is given by:ddxF(x)=f(b(x))\u22c5ddxb(x)\u2212f(a(x))\u22c5ddxa(x).\\frac{d}{dx} F(x) = f(b(x)) \\cdot \\frac{d}{dx} b(x) – f(a(x)) \\cdot \\frac{d}{dx} a(x).dxd\u200bF(x)=f(b(x))\u22c5dxd\u200bb(x)\u2212f(a(x))\u22c5dxd\u200ba(x).<\/p>\n\n\n\n

Step-by-Step Solution<\/h3>\n\n\n\n
    \n
  1. Identify the functions involved<\/strong>:\n
      \n
    • The integrand is ln\u2061(t)\\ln(t)ln(t).<\/li>\n\n\n\n
    • The limits of integration are a(x)=10xa(x) = 10xa(x)=10x and b(x)=29b(x) = 29b(x)=29.<\/li>\n<\/ul>\n<\/li>\n\n\n\n
    • Apply the formula<\/strong>:\n
        \n
      • The derivative of the upper limit b(x)=29b(x) = 29b(x)=29 with respect to xxx is 0 (since it is a constant).<\/li>\n\n\n\n
      • The derivative of the lower limit a(x)=10xa(x) = 10xa(x)=10x with respect to xxx is 10.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n

        Thus, by Part 2 of the Fundamental Theorem of Calculus, we get:ddx(\u222b10x29ln\u2061(t)\u2009dt)=ln\u2061(29)\u22c50\u2212ln\u2061(10x)\u22c510.\\frac{d}{dx} \\left( \\int_{10x}^{29} \\ln(t) \\, dt \\right) = \\ln(29) \\cdot 0 – \\ln(10x) \\cdot 10.dxd\u200b(\u222b10×29\u200bln(t)dt)=ln(29)\u22c50\u2212ln(10x)\u22c510.<\/p>\n\n\n\n

        Since the term involving ln\u2061(29)\\ln(29)ln(29) vanishes (because of the zero derivative of the upper limit), the result simplifies to:\u221210ln\u2061(10x).-10 \\ln(10x).\u221210ln(10x).<\/p>\n\n\n\n

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

        The derivative of the integral is:ddx(\u222b10x29ln\u2061(t)\u2009dt)=\u221210ln\u2061(10x).\\frac{d}{dx} \\left( \\int_{10x}^{29} \\ln(t) \\, dt \\right) = -10 \\ln(10x).dxd\u200b(\u222b10×29\u200bln(t)dt)=\u221210ln(10x).<\/p>\n\n\n\n

        This result follows directly from the application of Part 2 of the Fundamental Theorem of Calculus. We simply evaluated the integrand at the limits of integration and multiplied by the derivative of the respective limits.<\/p>\n\n\n\n

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

        ‘Calculate the derivative 10x In (t)dt dx 29 using Part 2 of the Fundamental Theorem of Calculus_ Enclose arguments of functions in parentheses_ For example; sin (2x) 10x In (t)dt = dx 29’ The Correct Answer and Explanation is: We are asked to calculate the derivative of the integral\u222b10x29ln\u2061(t)\u2009dtwith respect to\u2009x.\\int_{10x}^{29} \\ln(t) \\, dt \\quad \\text{with respect […]<\/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-263856","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/263856","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=263856"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/263856\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=263856"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=263856"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=263856"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}