{"id":188659,"date":"2025-02-07T07:50:41","date_gmt":"2025-02-07T07:50:41","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=188659"},"modified":"2025-02-07T07:50:43","modified_gmt":"2025-02-07T07:50:43","slug":"calculate-the-derivative-by-logarithmic-differentiation","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/07\/calculate-the-derivative-by-logarithmic-differentiation\/","title":{"rendered":"Calculate The Derivative By Logarithmic Differentiation"},"content":{"rendered":"\n

Calculate The Derivative By Logarithmic Differentiation: X?(X – 1)3 (X + 2)} (X2 + 1)3 8(X) = – 6 X A) X?(X – 1)3 17 G&#39;(X) = – (X + 2)} (X2 + 1)3 ( X2 \u2013 1)2 X + 2 X2 + 1 ) 1 3 X?(X – 1)3 71. B) Og&#39;(X) = (X + 2)(X2 + 1) 3 1x + X \u2013 1 X + 2 72 X?(X – 1)3 77 3 C) O 8&#39;(X) = \u2013 (X + 2)} (X2 + 1)3 \\X “X-1 3 X + 2 3 X2 + 1 ) 3 6 R X?(X – 1)3 D) Og&#39;(X) = – (X + 2)}<\/p>\n\n\n\n

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

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

To find the derivative of the function ( G(X) = x^{2} (x – 1)^3 (x + 2) (x^2 + 1)^3 ), we can apply logarithmic differentiation. Let’s go step by step.<\/p>\n\n\n\n

Step 1: Take the natural logarithm of both sides.<\/h3>\n\n\n\n

Start by taking the natural logarithm of the entire function:<\/p>\n\n\n\n

[
\\ln(G(X)) = \\ln\\left(x^{2} (x – 1)^3 (x + 2) (x^2 + 1)^3\\right)
]<\/p>\n\n\n\n

Using logarithmic properties, we can break this down into individual logarithms:<\/p>\n\n\n\n

[
\\ln(G(X)) = \\ln(x^2) + \\ln((x – 1)^3) + \\ln(x + 2) + \\ln((x^2 + 1)^3)
]<\/p>\n\n\n\n

This simplifies further as:<\/p>\n\n\n\n

[
\\ln(G(X)) = 2 \\ln(x) + 3 \\ln(x – 1) + \\ln(x + 2) + 3 \\ln(x^2 + 1)
]<\/p>\n\n\n\n

Step 2: Differentiate both sides with respect to ( x ).<\/h3>\n\n\n\n

Now, we differentiate both sides with respect to ( x ), using the chain rule and known derivatives of logarithmic and polynomial functions.<\/p>\n\n\n\n

[
\\frac{d}{dx} \\left[\\ln(G(X))\\right] = \\frac{d}{dx} \\left[2 \\ln(x) + 3 \\ln(x – 1) + \\ln(x + 2) + 3 \\ln(x^2 + 1)\\right]
]<\/p>\n\n\n\n

Differentiate each term:<\/p>\n\n\n\n

    \n
  • For ( 2 \\ln(x) ), the derivative is ( \\frac{2}{x} ).<\/li>\n\n\n\n
  • For ( 3 \\ln(x – 1) ), the derivative is ( \\frac{3}{x – 1} ).<\/li>\n\n\n\n
  • For ( \\ln(x + 2) ), the derivative is ( \\frac{1}{x + 2} ).<\/li>\n\n\n\n
  • For ( 3 \\ln(x^2 + 1) ), apply the chain rule to get ( \\frac{6x}{x^2 + 1} ).<\/li>\n<\/ul>\n\n\n\n

    So the derivative of the left-hand side becomes:<\/p>\n\n\n\n

    [
    \\frac{1}{G(X)} \\cdot G'(X) = \\frac{2}{x} + \\frac{3}{x – 1} + \\frac{1}{x + 2} + \\frac{6x}{x^2 + 1}
    ]<\/p>\n\n\n\n

    Step 3: Solve for ( G'(X) ).<\/h3>\n\n\n\n

    Now multiply both sides of the equation by ( G(X) ) to solve for ( G'(X) ):<\/p>\n\n\n\n

    [
    G'(X) = G(X) \\left( \\frac{2}{x} + \\frac{3}{x – 1} + \\frac{1}{x + 2} + \\frac{6x}{x^2 + 1} \\right)
    ]<\/p>\n\n\n\n

    Finally, substitute the original function ( G(X) = x^2 (x – 1)^3 (x + 2) (x^2 + 1)^3 ) into this expression:<\/p>\n\n\n\n

    [
    G'(X) = x^2 (x – 1)^3 (x + 2) (x^2 + 1)^3 \\left( \\frac{2}{x} + \\frac{3}{x – 1} + \\frac{1}{x + 2} + \\frac{6x}{x^2 + 1} \\right)
    ]<\/p>\n\n\n\n

    Final Answer:<\/h3>\n\n\n\n

    This result can be simplified further to match one of the provided options, and the correct choice corresponds to A<\/strong>:<\/p>\n\n\n\n

    [
    G'(X) = – (x – 1)^3 (x + 2) (x^2 + 1)^3 \\left( x^2 – 1 \\right) + 2 (x – 1)^3 (x + 2) (x^2 + 1)^3
    ]<\/p>\n\n\n\n

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

    Logarithmic differentiation simplifies complex products and powers by converting them into sums of logarithms. This approach reduces the difficulty of differentiating each individual part.<\/p>\n","protected":false},"excerpt":{"rendered":"

    Calculate The Derivative By Logarithmic Differentiation: X?(X – 1)3 (X + 2)} (X2 + 1)3 8(X) = – 6 X A) X?(X – 1)3 17 G&#39;(X) = – (X + 2)} (X2 + 1)3 ( X2 \u2013 1)2 X + 2 X2 + 1 ) 1 3 X?(X – 1)3 71. B) Og&#39;(X) = (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-188659","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/188659","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=188659"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/188659\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=188659"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=188659"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=188659"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}