{"id":5129,"date":"2021-09-08T20:34:50","date_gmt":"2021-09-08T15:04:50","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5129"},"modified":"2021-11-22T21:58:39","modified_gmt":"2021-11-22T16:28:39","slug":"differentiation-of-inverse-trigonometric-functions","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/differentiation-of-inverse-trigonometric-functions\/","title":{"rendered":"Differentiation of Inverse Trigonometric Functions"},"content":{"rendered":"

Here you will learn what is the differentiation of inverse trigonometric functions with examples.<\/p>\n

Let’s begin –<\/p>\n

Differentiation of Inverse Trigonometric Functions<\/h2>\n

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

\n

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

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

\n

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

(iii)<\/strong>  The differentiation of \\(tan^{-1}x\\)<\/strong> or arctanx<\/strong> with respect to x is \\(1\\over {1+x^2}\\)<\/strong>.<\/p>\n

\n

i.e. \\(d\\over dx\\) \\(tan^{-1}x\\) = \\(1\\over {1+x^2}\\).<\/p>\n<\/blockquote>\n

(iv)<\/strong>  The differentiation of \\(cot^{-1}x\\)<\/strong> or arccotx<\/strong> with respect to x is \\(-1\\over {1+x^2}\\)<\/strong>.<\/p>\n

\n

i.e. \\(d\\over dx\\) \\(cot^{-1}x\\) = \\(-1\\over {1+x^2}\\).<\/p>\n<\/blockquote>\n

(v)<\/strong>  If x \\(\\in\\) R – [-1, 1], then the differentiation of \\(sec^{-1}x\\)<\/strong> or arcsecx<\/strong> with respect to x is \\(1\\over |x|\\sqrt{x^2-1}\\)<\/strong>.<\/p>\n

\n

i.e. \\(d\\over dx\\) \\(sec^{-1}x\\) = \\(1\\over |x|\\sqrt{x^2-1}\\) , x \\(\\in\\) R – [-1, 1]<\/p>\n<\/blockquote>\n

(vi)<\/strong>  If x \\(\\in\\) R – [-1, 1], then the differentiation of \\(cosec^{-1}x\\)<\/strong> or arccosecx<\/strong> with respect to x is \\(-1\\over |x|\\sqrt{x^2-1}\\)<\/strong>.<\/p>\n

\n

i.e. \\(d\\over dx\\) \\(cosec^{-1}x\\) = \\(-1\\over |x|\\sqrt{x^2-1}\\) , x \\(\\in\\) R – [-1, 1]<\/p>\n<\/blockquote>\n

Example 1<\/span><\/strong> : find the differentiation of \\(sin^{-1}5x\\).<\/p>\n

Solution<\/span><\/strong> : Let y = \\(sin^{-1}5x\\)<\/p>\n

Now, \\(dy\\over dx\\) = \\(1\\over \\sqrt{1-(5x)^2}\\).5<\/p>\n

= \\(5\\over \\sqrt{1-25x^2}\\)<\/p>\n

Example 2<\/span><\/strong> : find the differentiation of \\(cos^{-1}5x\\).<\/p>\n

Solution<\/span><\/strong> : Let y = \\(cos^{-1}5x\\)<\/p>\n

Now, \\(dy\\over dx\\) = \\(-1\\over \\sqrt{1-(5x)^2}\\).5<\/p>\n

= \\(-5\\over \\sqrt{1-25x^2}\\)<\/p>\n

Example 3<\/span><\/strong> : find the differentiation of \\(tan^{-1}5x\\).<\/p>\n

Solution<\/span><\/strong> : Let y = \\(tan^{-1}5x\\)<\/p>\n

Now, \\(dy\\over dx\\) = \\(1\\over {1+(5x)^2}\\).5<\/p>\n

= \\(5\\over {1+25x^2}\\)<\/p>\n

Example 4<\/span><\/strong> : find the differentiation of \\(sec^{-1}5x\\).<\/p>\n

Solution<\/span><\/strong> : Let y = \\(sec^{-1}5x\\)<\/p>\n

Now, \\(dy\\over dx\\) = \\(1\\over |5x|\\sqrt{(5x)^2 – 1}\\).5<\/p>\n

= \\(5\\over |5x|\\sqrt{25x^2-1}\\) = \\(1\\over x\\sqrt{25x^2-1}\\)<\/p>\n

Example 5<\/span><\/strong> : find the differentiation of \\(cosec^{-1}5x\\).<\/p>\n

Solution<\/span><\/strong> : Let y = \\(cosec^{-1}5x\\)<\/p>\n

Now, \\(dy\\over dx\\) = \\(-1\\over |5x|\\sqrt{1-(5x)^2}\\).5<\/p>\n

= \\(-5\\over |5x|\\sqrt{25x^2-1}\\) = \\(-1\\over x\\sqrt{25x^2-1}\\)<\/p>\n\n\n

\n
Next -Differentiation of Implicit Function<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Chain Rule in Differentiation with Examples<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn what is the differentiation of inverse trigonometric functions with examples. Let’s begin – Differentiation of Inverse Trigonometric Functions (i)  If x \\(\\in\\) (-1, 1), then the differentiation of \\(sin^{-1}x\\) or arcsinx with respect to x is \\(1\\over \\sqrt{1-x^2}\\). i.e. \\(d\\over dx\\) \\(sin^{-1}x\\) = \\(1\\over \\sqrt{1-x^2}\\) , for x \\(\\in\\) (-1, 1). …<\/p>\n

Differentiation of Inverse Trigonometric Functions<\/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":[324,275,323],"yoast_head":"\nDifferentiation of Inverse Trigonometric Functions - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is the differentiation of inverse trigonometric functions with examples.\" \/>\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-inverse-trigonometric-functions\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Differentiation of Inverse Trigonometric Functions - 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