{"id":5527,"date":"2021-09-16T20:28:21","date_gmt":"2021-09-16T14:58:21","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5527"},"modified":"2021-11-22T17:34:06","modified_gmt":"2021-11-22T12:04:06","slug":"differentiation-of-cos-inverse-x","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/differentiation-of-cos-inverse-x\/","title":{"rendered":"Differentiation of cos inverse x"},"content":{"rendered":"

Here you will learn differentiation of cos inverse x or arccos x by using chain rule.<\/p>\n

Let’s begin –<\/p>\n

Differentiation of cos inverse x or \\(cos^{-1}x\\) :<\/h2>\n
\n

If x \\(\\in\\) (-1, 1) , then the differentiation of \\(cos^{-1}x\\) with respect to x is \\(-1\\over \\sqrt{1 – x^2}\\).<\/p>\n

i.e. \\(d\\over dx\\) \\(cos^{-1}x\\) = \\(-1\\over \\sqrt{1 – x^2}\\) , for x \\(\\in\\) (-1, 1).<\/p>\n<\/blockquote>\n

Proof using chain rule :<\/h2>\n
\n

Let y = \\(cos^{-1}x\\). Then,<\/p>\n

\\(cos(cos^{-1}x)\\) = x<\/p>\n

\\(\\implies\\) cos y = x<\/p>\n

Differentiating both sides with respect to x, we get<\/p>\n

\\(d\\over dx\\)(cos y) = \\(d\\over dx\\)(x)<\/p>\n

\\(d\\over dx\\) (cos y) = 1<\/p>\n

By chain rule,<\/p>\n

-sin y \\(dy\\over dx\\) = 1<\/p>\n

\\(dy\\over dx\\) = \\(-1\\over sin y\\)<\/p>\n

\\(dy\\over dx\\) = \\(-1\\over \\sqrt{1 – cos^2 y}\\)<\/p>\n

\\(\\implies\\) \\(dy\\over dx\\) = \\(-1\\over \\sqrt{1 – x^2}\\)<\/p>\n

\\(\\implies\\) \\(d\\over dx\\) \\(cos^{-1}x\\) = \\(-1\\over \\sqrt{1 – x^2}\\)\u00a0<\/p>\n

Hence, the differentiation of \\(cos^{-1}x\\) with respect to x is \\(-1\\over \\sqrt{1 – x^2}\\).<\/p>\n<\/blockquote>\n

Example<\/strong><\/span> : What is the differentiation of \\(cos^{-1} x^3\\) with respect to x ?<\/p>\n

Solution<\/strong><\/span> : Let y = \\(cos^{-1} x^3\\)<\/p>\n

Differentiating both sides with respect to x and using chain rule, we get<\/p>\n

\\(dy\\over dx\\) = \\(d\\over dx\\) (\\(cos^{-1} x^3\\))<\/p>\n

\\(dy\\over dx\\) = \\(-1\\over \\sqrt{1 – x^6}\\).\\(3x^2\\) = \\(-3x^2\\over \\sqrt{1 – x^6}\\)<\/p>\n

Hence, \\(d\\over dx\\) (\\(cos^{-1} x^3\\)) = \\(-3x^2\\over \\sqrt{1 – x^6}\\)<\/p>\n

Example<\/strong><\/span> : What is the differentiation of x + \\(cos^{-1} x\\) with respect to x ?<\/p>\n

Solution<\/strong><\/span> : Let y = x + \\(cos^{-1} x\\)<\/p>\n

Differentiating both sides with respect to x, we get<\/p>\n

\\(dy\\over dx\\) = \\(d\\over dx\\) (x) + \\(d\\over dx\\) (\\(cos^{-1} x\\))<\/p>\n

\\(dy\\over dx\\) = 1 + \\(-1\\over \\sqrt{1 – x^2}\\)<\/p>\n

Hence, \\(d\\over dx\\) (x + \\(cos^{-1} x\\)) = 1 – \\(1\\over \\sqrt{1 – x^2}\\)<\/p>\n

\n

Related Questions<\/h3>\n

What is the Differentiation of cos x ?<\/a><\/p>\n

What is the Integration of cos Inverse x ?<\/a><\/p>\n

What is the Differentiation of cosec inverse x ?<\/a><\/p>\n\n\n

\n
Next – Differentiation of tan inverse x<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Differentiation of sin inverse x<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn differentiation of cos inverse x or arccos x by using chain rule. Let’s begin – Differentiation of cos inverse x or \\(cos^{-1}x\\) : If x \\(\\in\\) (-1, 1) , then the differentiation of \\(cos^{-1}x\\) with respect to x is \\(-1\\over \\sqrt{1 – x^2}\\). i.e. \\(d\\over dx\\) \\(cos^{-1}x\\) = \\(-1\\over \\sqrt{1 – …<\/p>\n

Differentiation of cos inverse x<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","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":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[36],"tags":[282,281,275,280],"yoast_head":"\nDifferentiation of cos inverse x - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn differentiation of cos inverse x or arccos x by using chain rule.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/differentiation-of-cos-inverse-x\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Differentiation of cos inverse x - 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