{"id":269957,"date":"2025-07-24T22:34:23","date_gmt":"2025-07-24T22:34:23","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=269957"},"modified":"2025-07-24T22:34:26","modified_gmt":"2025-07-24T22:34:26","slug":"the-derivative-of-the-function-y-cot-1vzx-is-202v2x3-26-1v2x-3","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/24\/the-derivative-of-the-function-y-cot-1vzx-is-202v2x3-26-1v2x-3\/","title":{"rendered":"The derivative of the function y = cot^(-1)(vZx) is: (20+2)v(2x+3) (26-1)v(2x-3)"},"content":{"rendered":"\n

The derivative of the function y = cot^(-1)(vZx) is: (20+2)v(2x+3) (26-1)v(2x-3) This option This option y’ y’ (20+2)v(2x+3) (26-1)v(2x-3) This option This option
The derivative of the function y = cot^(-1)(vZx) is: (20+2)v(2x+3) (26-1)v(2x-3) This option This option y’ y’ (20+2)v(2x+3) (26-1)v(2x-3) This option This option<\/p>\n\n\n\n

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

To find the derivative of y=cot\u2061\u22121(vZx)y = \\cot^{-1}(vZx)y=cot\u22121(vZx), where vZxvZxvZx is the expression inside the inverse cotangent function, we need to apply the chain rule and the derivative formula for the inverse cotangent function.<\/p>\n\n\n\n

Step 1: Apply the derivative rule for cot\u2061\u22121(x)\\cot^{-1}(x)cot\u22121(x)<\/h3>\n\n\n\n

The derivative of the inverse cotangent function, cot\u2061\u22121(x)\\cot^{-1}(x)cot\u22121(x), is given by:ddxcot\u2061\u22121(x)=\u221211+x2\\frac{d}{dx} \\cot^{-1}(x) = -\\frac{1}{1 + x^2}dxd\u200bcot\u22121(x)=\u22121+x21\u200b<\/p>\n\n\n\n

Now, in your case, the function inside the inverse cotangent is vZxvZxvZx. So, the derivative of y=cot\u2061\u22121(vZx)y = \\cot^{-1}(vZx)y=cot\u22121(vZx) with respect to xxx is:y\u2032=\u2212ddx[cot\u2061\u22121(vZx)]y’ = -\\frac{d}{dx} \\left[ \\cot^{-1}(vZx) \\right]y\u2032=\u2212dxd\u200b[cot\u22121(vZx)]<\/p>\n\n\n\n

Step 2: Apply the chain rule<\/h3>\n\n\n\n

We need to apply the chain rule because vZxvZxvZx is a function of xxx. To differentiate it, we first find the derivative of the inside function vZxvZxvZx with respect to xxx. If vZxvZxvZx is of the form v\u22c5Zxv \\cdot Zxv\u22c5Zx, then the derivative of vZxvZxvZx with respect to xxx is simply vZvZvZ.<\/p>\n\n\n\n

Thus, applying the chain rule:y\u2032=\u221211+(vZx)2\u22c5ddx(vZx)y’ = -\\frac{1}{1 + (vZx)^2} \\cdot \\frac{d}{dx}(vZx)y\u2032=\u22121+(vZx)21\u200b\u22c5dxd\u200b(vZx)<\/p>\n\n\n\n

The derivative of vZxvZxvZx with respect to xxx is vZvZvZ, so:y\u2032=\u2212vZ1+(vZx)2y’ = -\\frac{vZ}{1 + (vZx)^2}y\u2032=\u22121+(vZx)2vZ\u200b<\/p>\n\n\n\n

Step 3: Simplify and Interpret the Options<\/h3>\n\n\n\n

You mentioned several options like:<\/p>\n\n\n\n

    \n
  • (20+2)v(2x+3)<\/li>\n\n\n\n
  • (26-1)v(2x-3)<\/li>\n<\/ul>\n\n\n\n

    These options don’t match the standard form of the derivative for cot\u2061\u22121(vZx)\\cot^{-1}(vZx)cot\u22121(vZx), as they involve linear terms in xxx and constants that don’t appear in the general derivative formula for inverse cotangent. It’s likely that there is a specific context or additional information about the function vZxvZxvZx that has been omitted. However, based on the standard derivative rules, the correct expression for the derivative would be:y\u2032=\u2212vZ1+(vZx)2y’ = -\\frac{vZ}{1 + (vZx)^2}y\u2032=\u22121+(vZx)2vZ\u200b<\/p>\n\n\n\n

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

    The derivative of the function y = cot^(-1)(vZx) is: (20+2)v(2x+3) (26-1)v(2x-3) This option This option y’ y’ (20+2)v(2x+3) (26-1)v(2x-3) This option This optionThe derivative of the function y = cot^(-1)(vZx) is: (20+2)v(2x+3) (26-1)v(2x-3) This option This option y’ y’ (20+2)v(2x+3) (26-1)v(2x-3) This option This option The Correct Answer and Explanation is: To find the derivative […]<\/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-269957","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/269957","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=269957"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/269957\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=269957"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=269957"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=269957"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}