{"id":390,"date":"2023-07-03T09:49:11","date_gmt":"2023-07-03T09:49:11","guid":{"rendered":"https:\/\/mathority.org\/id\/turunan-dari-tangen-busur-hiperbolik\/"},"modified":"2023-07-03T09:49:11","modified_gmt":"2023-07-03T09:49:11","slug":"turunan-dari-tangen-busur-hiperbolik","status":"publish","type":"post","link":"https:\/\/mathority.org\/id\/turunan-dari-tangen-busur-hiperbolik\/","title":{"rendered":"Turunan dari tangen busur hiperbolik"},"content":{"rendered":"<p>Di sini Anda akan menemukan cara menurunkan tangen hiperbolik suatu fungsi. Anda juga akan dapat melihat contoh penyelesaian turunan trigonometri jenis ini dan, terakhir, kami akan menunjukkan rumus turunan tangen busur hiperbolik. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-de-la-derivada-de-la-arcotangente-hiperbolica\"><\/span> Rumus turunan dari tangen busur hiperbolik<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>Turunan dari tangen busur hiperbolik x adalah satu per satu dikurangi x kuadrat.<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-33ae63a662489900a94430ce0dac1b60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arctanh}(x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{1}{1-x^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"413\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Oleh karena itu, <strong>turunan tangen busur hiperbolik suatu fungsi<\/strong> sama dengan hasil bagi turunan fungsi tersebut dibagi satu dikurangi kuadrat fungsi tersebut.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3d5c743ab52bf834518230f3446aaa9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arctanh}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{u'}{1-u^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"413\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Sebenarnya kedua rumus tersebut sama, namun pada rumus kedua diterapkan aturan rantai. Misalnya, mengganti x dengan u memberi kita rumus pertama karena turunan dari x adalah 1. <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/derivee-de-larctangente-hyperbolique.webp\" alt=\"turunan dari tangen busur hiperbolik\" class=\"wp-image-2343\" width=\"393\" height=\"298\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Sama seperti tangen busur yang merupakan kebalikan dari fungsi tangen, maka tangen busur hiperbolik adalah kebalikan dari tangen hiperbolik. Meski begitu turunannya sangat berbeda, turunan fungsi trigonometri ini bisa kamu cek di sini:<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Lihat:<\/strong> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/id\/turunan-dari-garis-singgung-hiperbolik\/\">rumus turunan tangen hiperbolik<\/a><\/span> <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplos-de-la-derivada-de-la-arcotangente-hiperbolica\"><\/span> Contoh turunan dari arctangen hiperbolik<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Contoh 1<\/h3>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ad1cd9320973ca2c5d2b83434086f629_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arctanh}(2x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Logikanya, kita harus menerapkan aturan turunan dari tangen busur hiperbolik:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3d5c743ab52bf834518230f3446aaa9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arctanh}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{u'}{1-u^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"413\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Turunan dari 2x adalah 2, jadi masukkan dua pada pembilang pecahan dan satu dikurangi 2x kuadrat pada penyebutnya:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9820d63e99b4b29c41d6fd14a3426815_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arctanh}(2x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{2}{1-(2x)^2}}=\\cfrac{2}{1- 4x^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"528\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"> Contoh 2<\/h3>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3e4e12eb6cf782403fe0de4f37bc025f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arctanh}(e^{3x})\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"154\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Untuk menyelesaikan turunan fungsi ini, kita perlu menggunakan rumus turunan tangen busur hiperbolik.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3d5c743ab52bf834518230f3446aaa9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arctanh}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{u'}{1-u^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"413\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Selain itu, fungsi argumen tangen busur hiperbolik adalah fungsi gabungan, jadi kita juga perlu menerapkan aturan rantai: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0eb0da6a9477e040476051a829238c84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arctanh}(e^{3x}) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{3\\cdot e^{3x}}{1-\\left(e^{3x}\\right)^2}=\\cfrac{3e^{3x}}{1-3^{6x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"534\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"demostracion-de-la-derivada-de-la-arcotangente-hiperbolica\"><\/span>Bukti turunan dari tangen busur hiperbolik<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Pada bagian akhir ini, kita akan mendemonstrasikan rumus turunan tangen busur hiperbolik.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-17261fa2031302bfad1883eb39b7116d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=\\text{arctanh}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"115\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Karena tangen busur hiperbolik adalah kebalikan dari tangen hiperbolik, kita dapat menyatakan persamaan sebelumnya dengan cara lain:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-27ba9a49fdc790b3131113b5ae592e2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=\\text{tanh}(y)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"91\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Sekarang kita bedakan kedua ruas persamaan:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4f8879b6d2f8df36bd6f3e5c817f21cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"1=\\cfrac{1}{\\text{cosh}^2(y)}\\cdot y'\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"124\" style=\"vertical-align: -19px;\"><\/p>\n<\/p>\n<p> Kami membersihkan Anda:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3bb09e461662267f0cbab52cf6e0bcac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y'=\\text{cosh}^2(y)\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"101\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Sebaliknya, kita mengetahui bahwa selisih kuadrat kosinus hiperbolik dan sinus hiperbolik menghasilkan 1. Oleh karena itu, kita dapat mengubah ekspresi sebelumnya menjadi pecahan:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e726904c011eb3ab9ff264426988d029_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{cosh}^2(y)-\\text{senh}^2(y)=1\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"183\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1820ba4560d8d0109af605b6e2757c93_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y'=\\cfrac{\\text{cosh}^2(y)}{1}=\\cfrac{\\text{cosh}^2(y)}{\\text{cosh}^2(y)-\\text{senh}^2(y)}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"281\" style=\"vertical-align: -19px;\"><\/p>\n<\/p>\n<p> Kami membagi semua suku pecahan dengan kuadrat kosinus hiperbolik:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9cc80abe73f130f4ec1c39cbf5d7e8ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y'=\\cfrac{\\cfrac{\\text{cosh}^2(y)}{\\text{cosh}^2(y)}}{\\cfrac{\\text{cosh}^2(y)}{\\text{cosh}^2(y)}-\\cfrac{\\text{senh}^2(y)}{\\text{cosh}^2(y)}}\" title=\"Rendered by QuickLaTeX.com\" height=\"101\" width=\"195\" style=\"vertical-align: -46px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b150bbe95858decf7312b869b95d24b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y'=\\cfrac{1}{1-\\cfrac{\\text{senh}^2(y)}{\\text{cosh}^2(y)}}\" title=\"Rendered by QuickLaTeX.com\" height=\"72\" width=\"138\" style=\"vertical-align: -46px;\"><\/p>\n<\/p>\n<p> Hasil bagi sinus hiperbolik antara kosinus hiperbolik sama dengan tangen hiperbolik, oleh karena itu:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-12f286528bc0635705aadbe510b6ceb7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tanh}(x)=\\cfrac{\\text{senh}(x)}{\\text{cosh}(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"144\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-725424805ce03fcabd470e9448c91f2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y'=\\cfrac{1}{1-\\text{tanh}^2(y)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"136\" style=\"vertical-align: -19px;\"><\/p>\n<\/p>\n<p> Namun, seperti yang kita lihat di awal pembuktian, garis singgung hiperbolik setara dengan variabel x, oleh karena itu kita dapat mensubstitusikan ekspresi tersebut sehingga memperoleh rumus turunan dari garis singgung busur hiperbolik: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b90ecc88a8cbc7f110840727da48e632_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y'=\\cfrac{1}{1-x^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"88\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"articulos-relacionados\"><\/span> Barang Serupa<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul>\n<li> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/id\/turunan-dari-kotangen-hiperbolik\/\">Rumus turunan kotangen hiperbolik<\/a><\/span><\/li>\n<li> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/id\/turunan-dari-arctangen\/\">rumus turunan arckotangen<\/a><\/span><\/li>\n<li> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/id\/turunan-dari-arctangent-1\/\">Rumus turunan arctangen<\/a><\/span><\/li>\n<li><span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/id\/turunan-dari-kotangen\/\">Rumus turunan kotangen<\/a><\/span><\/li>\n<li> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/id\/turunan-dari-garis-singgung\/\">Rumus turunan garis singgung<\/a><\/span><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Di sini Anda akan menemukan cara menurunkan tangen hiperbolik suatu fungsi. Anda juga akan dapat melihat contoh penyelesaian turunan trigonometri jenis ini dan, terakhir, kami akan menunjukkan rumus turunan tangen busur hiperbolik. Rumus turunan dari tangen busur hiperbolik Turunan dari tangen busur hiperbolik x adalah satu per satu dikurangi x kuadrat. Oleh karena itu, turunan &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/id\/turunan-dari-tangen-busur-hiperbolik\/\"> <span class=\"screen-reader-text\">Turunan dari tangen busur hiperbolik<\/span> Selengkapnya &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","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":"","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":"","footnotes":""},"categories":[38],"tags":[],"class_list":["post-390","post","type-post","status-publish","format-standard","hentry","category-derivatif"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Turunan dari tangen busur hiperbolik - Mathority<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathority.org\/id\/turunan-dari-tangen-busur-hiperbolik\/\" \/>\n<meta property=\"og:locale\" content=\"id_ID\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Turunan dari tangen busur hiperbolik - Mathority\" \/>\n<meta property=\"og:description\" content=\"Di sini Anda akan menemukan cara menurunkan tangen hiperbolik suatu fungsi. Anda juga akan dapat melihat contoh penyelesaian turunan trigonometri jenis ini dan, terakhir, kami akan menunjukkan rumus turunan tangen busur hiperbolik. Rumus turunan dari tangen busur hiperbolik Turunan dari tangen busur hiperbolik x adalah satu per satu dikurangi x kuadrat. 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