To find the derivative dydx\\frac{dy}{dx}dxdy\u200b of the given function y=tan\u2061\u22121(x2+1)y = \\tan^{-1}(x^2 + 1)y=tan\u22121(x2+1), we’ll need to use implicit differentiation and the chain rule.<\/p>\n\n\n\n
Step 1: Differentiation of the Inverse Tangent Function<\/h3>\n\n\n\n
The derivative of the inverse tangent function tan\u2061\u22121(u)\\tan^{-1}(u)tan\u22121(u), where uuu is a function of xxx, is: ddx(tan\u2061\u22121(u))=11+u2\u22c5dudx\\frac{d}{dx}\\left(\\tan^{-1}(u)\\right) = \\frac{1}{1 + u^2} \\cdot \\frac{du}{dx}dxd\u200b(tan\u22121(u))=1+u21\u200b\u22c5dxdu\u200b<\/p>\n\n\n\n
In this case, u=x2+1u = x^2 + 1u=x2+1, so we need to differentiate u=x2+1u = x^2 + 1u=x2+1 with respect to xxx.<\/p>\n\n\n\n
Step 2: Differentiate the Inner Function<\/h3>\n\n\n\n
The derivative of u=x2+1u = x^2 + 1u=x2+1 with respect to xxx is: dudx=2x\\frac{du}{dx} = 2xdxdu\u200b=2x<\/p>\n\n\n\n
Step 3: Apply the Chain Rule<\/h3>\n\n\n\n
Now we can apply the chain rule to differentiate the entire expression. We get: dydx=ddx(tan\u2061\u22121(x2+1))=11+(x2+1)2\u22c52x\\frac{dy}{dx} = \\frac{d}{dx}\\left(\\tan^{-1}(x^2 + 1)\\right) = \\frac{1}{1 + (x^2 + 1)^2} \\cdot 2xdxdy\u200b=dxd\u200b(tan\u22121(x2+1))=1+(x2+1)21\u200b\u22c52x<\/p>\n\n\n\n
Step 4: Simplify the Expression<\/h3>\n\n\n\n
First, simplify the denominator: 1+(x2+1)2=1+(x4+2×2+1)=x4+2×2+21 + (x^2 + 1)^2 = 1 + (x^4 + 2x^2 + 1) = x^4 + 2x^2 + 21+(x2+1)2=1+(x4+2×2+1)=x4+2×2+2<\/p>\n\n\n\n
Now, substitute this back into the expression for dydx\\frac{dy}{dx}dxdy\u200b: dydx=2xx4+2×2+2\\frac{dy}{dx} = \\frac{2x}{x^4 + 2x^2 + 2}dxdy\u200b=x4+2×2+22x\u200b<\/p>\n\n\n\n
Final Answer:<\/h3>\n\n\n\n
dydx=2xx4+2×2+2\\frac{dy}{dx} = \\frac{2x}{x^4 + 2x^2 + 2}dxdy\u200b=x4+2×2+22x\u200b<\/p>\n\n\n\n
Explanation:<\/h3>\n\n\n\n
The derivative dydx\\frac{dy}{dx}dxdy\u200b is found by applying the chain rule to the inverse tangent function. First, differentiate the outer function, tan\u2061\u22121(u)\\tan^{-1}(u)tan\u22121(u), where u=x2+1u = x^2 + 1u=x2+1, then multiply by the derivative of the inner function u=x2+1u = x^2 + 1u=x2+1, which is 2x2x2x. The denominator 1+(x2+1)21 + (x^2 + 1)^21+(x2+1)2 simplifies to x4+2×2+2x^4 + 2x^2 + 2×4+2×2+2.<\/p>\n\n\n\n (0 Find the derivative dy dx tan-1 x2 y = x2 + 1 The Correct Answer and Explanation is: To find the derivative dydx\\frac{dy}{dx}dxdy\u200b of the given function y=tan\u2061\u22121(x2+1)y = \\tan^{-1}(x^2 + 1)y=tan\u22121(x2+1), we’ll need to use implicit differentiation and the chain rule. Step 1: Differentiation of the Inverse Tangent Function The derivative of the […]<\/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-270770","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/270770","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=270770"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/270770\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=270770"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=270770"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=270770"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}![\"\"](\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner6-1984.jpeg\")
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