{"id":182098,"date":"2025-01-13T10:19:14","date_gmt":"2025-01-13T10:19:14","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=182098"},"modified":"2025-01-13T10:19:16","modified_gmt":"2025-01-13T10:19:16","slug":"use-implicit-differentiation-to-find-dy-dx-and-d2y-dx2","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/13\/use-implicit-differentiation-to-find-dy-dx-and-d2y-dx2\/","title":{"rendered":"Use Implicit Differentiation To Find Dy\/Dx And D2y\/Dx2"},"content":{"rendered":"\n

Use Implicit Differentiation To Find Dy\/Dx And D2y\/Dx2<\/p>\n\n\n\n

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

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

Implicit differentiation is a technique used to find derivatives of functions defined implicitly, where ( y ) is a function of ( x ) but is not explicitly solved for ( y ). This method is particularly useful when dealing with equations that are difficult or impossible to solve explicitly for ( y ).<\/p>\n\n\n\n

Finding ( \\frac{dy}{dx} ):<\/strong><\/p>\n\n\n\n

Consider the equation ( x^2 + y^2 = 25 ), which represents a circle. To find ( \\frac{dy}{dx} ), follow these steps:<\/p>\n\n\n\n

    \n
  1. Differentiate both sides with respect to ( x ):<\/strong>
    [
    \\frac{d}{dx}(x^2 + y^2) = \\frac{d}{dx}(25)
    ]
    Applying the chain rule to ( y^2 ) (since ( y ) is a function of ( x )):
    [
    2x + 2y \\cdot \\frac{dy}{dx} = 0
    ]<\/li>\n\n\n\n
  2. Solve for ( \\frac{dy}{dx} ):<\/strong>
    [
    2y \\cdot \\frac{dy}{dx} = -2x
    ]
    [
    \\frac{dy}{dx} = -\\frac{x}{y}
    ]<\/li>\n<\/ol>\n\n\n\n

    Thus, the derivative of ( y ) with respect to ( x ) is ( \\frac{dy}{dx} = -\\frac{x}{y} ).<\/p>\n\n\n\n

    Finding ( \\frac{d^2y}{dx^2} ):<\/strong><\/p>\n\n\n\n

    To find the second derivative, differentiate ( \\frac{dy}{dx} = -\\frac{x}{y} ) with respect to ( x ):<\/p>\n\n\n\n

      \n
    1. Differentiate ( \\frac{dy}{dx} = -\\frac{x}{y} ) with respect to ( x ):<\/strong>
      [
      \\frac{d}{dx}\\left(-\\frac{x}{y}\\right) = \\frac{d}{dx}\\left(\\frac{dy}{dx}\\right)
      ]
      Using the quotient rule:
      [
      \\frac{d^2y}{dx^2} = \\frac{(-y)(1) – (-x)\\left(\\frac{dy}{dx}\\right)}{y^2}
      ]
      Substitute ( \\frac{dy}{dx} = -\\frac{x}{y} ):
      [
      \\frac{d^2y}{dx^2} = \\frac{-y + x \\cdot \\left(-\\frac{x}{y}\\right)}{y^2}
      ]
      Simplify the numerator:
      [
      \\frac{d^2y}{dx^2} = \\frac{-y – \\frac{x^2}{y}}{y^2}
      ]
      Combine terms in the numerator:
      [
      \\frac{d^2y}{dx^2} = \\frac{-y^2 – x^2}{y^3}
      ]
      Since ( x^2 + y^2 = 25 ), substitute this into the numerator:
      [
      \\frac{d^2y}{dx^2} = \\frac{-25}{y^3}
      ]<\/li>\n<\/ol>\n\n\n\n

      Therefore, the second derivative of ( y ) with respect to ( x ) is ( \\frac{d^2y}{dx^2} = -\\frac{25}{y^3} ).<\/p>\n\n\n\n

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

      Implicit differentiation allows us to differentiate equations where ( y ) is not explicitly expressed as a function of ( x ). By differentiating both sides of the equation with respect to ( x ) and applying the chain rule, we can find ( \\frac{dy}{dx} ). To find higher-order derivatives like ( \\frac{d^2y}{dx^2} ), we differentiate ( \\frac{dy}{dx} ) with respect to ( x ) again, applying the chain rule as needed. This method is particularly useful for curves defined by equations that are difficult to solve explicitly for ( y ).<\/p>\n","protected":false},"excerpt":{"rendered":"

      Use Implicit Differentiation To Find Dy\/Dx And D2y\/Dx2 The Correct Answer and Explanation is : Implicit differentiation is a technique used to find derivatives of functions defined implicitly, where ( y ) is a function of ( x ) but is not explicitly solved for ( y ). This method is particularly useful when dealing […]<\/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-182098","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/182098","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=182098"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/182098\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=182098"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=182098"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=182098"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}