{"id":227704,"date":"2025-06-06T12:02:56","date_gmt":"2025-06-06T12:02:56","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=227704"},"modified":"2025-06-06T12:02:58","modified_gmt":"2025-06-06T12:02:58","slug":"determine-whether-each-statement-is-true-or-false","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/06\/determine-whether-each-statement-is-true-or-false\/","title":{"rendered":"Determine whether each statement is true or false."},"content":{"rendered":"\n

Determine whether each statement is true or false. You have one submission for each statement. (a) The derivative of
is equal to
. \\OTrue \\OFalse (b) The derivative of
is equal to
. \\OTrue \\OFalse<\/p>\n\n\n\n

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

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

Let’s analyze each part of the question in the image.<\/p>\n\n\n\n

\n\n\n\n

(a) The derivative of e3xe^{3x} is equal to e3xe^{3x}<\/strong><\/h3>\n\n\n\n

Answer: False<\/strong><\/p>\n\n\n\n

Explanation:<\/strong><\/p>\n\n\n\n

To find the derivative of e3xe^{3x}, we must apply the chain rule<\/strong> from calculus. The chain rule tells us that if a function is composed of an outer function and an inner function, say f(g(x))f(g(x)), then the derivative is: ddxf(g(x))=f\u2032(g(x))\u22c5g\u2032(x)\\frac{d}{dx}f(g(x)) = f'(g(x)) \\cdot g'(x)<\/p>\n\n\n\n

In this case:<\/p>\n\n\n\n

    \n
  • Outer function: eue^u<\/li>\n\n\n\n
  • Inner function: u=3xu = 3x<\/li>\n<\/ul>\n\n\n\n

    So, the derivative is: ddxe3x=e3x\u22c5ddx(3x)=e3x\u22c53=3e3x\\frac{d}{dx}e^{3x} = e^{3x} \\cdot \\frac{d}{dx}(3x) = e^{3x} \\cdot 3 = 3e^{3x}<\/p>\n\n\n\n

    Therefore, ddx(e3x)\u2260e3x\\frac{d}{dx}(e^{3x}) \\neq e^{3x}, and the statement is False<\/strong>.<\/p>\n\n\n\n

    \n\n\n\n

    (b) The derivative of exf(x)e^x f(x) is equal to ex[f(x)+f\u2032(x)]e^x[f(x) + f'(x)]<\/strong><\/h3>\n\n\n\n

    Answer: True<\/strong><\/p>\n\n\n\n

    Explanation:<\/strong><\/p>\n\n\n\n

    This expression requires the use of the product rule<\/strong>, which states: ddx(u\u22c5v)=u\u2032v+uv\u2032\\frac{d}{dx}(u \\cdot v) = u’v + uv’<\/p>\n\n\n\n

    Here:<\/p>\n\n\n\n

      \n
    • u=exu = e^x and u\u2032=exu’ = e^x<\/li>\n\n\n\n
    • v=f(x)v = f(x) and v\u2032=f\u2032(x)v’ = f'(x)<\/li>\n<\/ul>\n\n\n\n

      Apply the product rule: ddx(exf(x))=ex\u22c5f(x)+ex\u22c5f\u2032(x)\\frac{d}{dx}(e^x f(x)) = e^x \\cdot f(x) + e^x \\cdot f'(x)<\/p>\n\n\n\n

      Factor out exe^x: =ex[f(x)+f\u2032(x)]= e^x[f(x) + f'(x)]<\/p>\n\n\n\n

      So, the derivative is indeed ex[f(x)+f\u2032(x)]e^x[f(x) + f'(x)], and the statement is True<\/strong>.<\/p>\n\n\n\n

      \n\n\n\n

      Summary:<\/strong><\/h3>\n\n\n\n
        \n
      • (a): False<\/strong> \u2013 because ddx(e3x)=3e3x\\frac{d}{dx}(e^{3x}) = 3e^{3x}<\/li>\n\n\n\n
      • (b): True<\/strong> \u2013 due to correct application of the product rule<\/li>\n<\/ul>\n\n\n\n

        These questions test understanding of fundamental differentiation rules: the chain rule and the product rule, both essential in calculus.<\/p>\n\n\n\n

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

        Determine whether each statement is true or false. You have one submission for each statement. (a) The derivative ofis equal to. \\OTrue \\OFalse (b) The derivative ofis equal to. \\OTrue \\OFalse The Correct Answer and Explanation is: Let’s analyze each part of the question in the image. (a) The derivative of e3xe^{3x} is equal to […]<\/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-227704","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/227704","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=227704"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/227704\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=227704"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=227704"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=227704"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}