{"id":37,"date":"2023-09-17T11:00:07","date_gmt":"2023-09-17T11:00:07","guid":{"rendered":"https:\/\/mathority.org\/id\/rantai-aturan-turunan\/"},"modified":"2023-09-17T11:00:07","modified_gmt":"2023-09-17T11:00:07","slug":"rantai-aturan-turunan","status":"publish","type":"post","link":"https:\/\/mathority.org\/id\/rantai-aturan-turunan\/","title":{"rendered":"Aturan rantai (turunan)"},"content":{"rendered":"<p>Di sini Anda akan mempelajari apa itu aturan rantai dan cara menurunkan fungsi menggunakan aturan rantai. Selain itu, Anda akan dapat melihat beberapa contoh turunan yang diselesaikan dengan aturan rantai dan Anda bahkan dapat berlatih dengan latihan penyelesaian langkah demi langkah pada turunan yang menerapkan aturan rantai. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-es-la-regla-de-la-cadena\"><\/span> Apa aturan rantainya?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>Aturan rantai adalah rumus yang digunakan untuk menurunkan fungsi komposit.<\/strong> Aturan rantai menyatakan bahwa turunan suatu fungsi komposit <em>f(g(x))<\/em> sama dengan turunan <em>f'(g(x))<\/em> dikalikan dengan turunan <em>g'(x)<\/em> . <\/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\/regle-de-la-chaine.webp\" alt=\"aturan rantai\" class=\"wp-image-2207\" width=\"269\" height=\"269\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Lihat:<\/strong> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/id\/komposisi-fungsi-fungsi-komposit\/\">fungsi komposit<\/a><\/span><\/p>\n<p> Secara informal, aturan rantai sering dikatakan untuk <em>mendiferensiasikan suatu fungsi kemudian mengalikannya dengan apa yang ada di dalamnya<\/em> .<\/p>\n<p> Rumus aturan rantai memungkinkan kita untuk membedakan fungsi majemuk dengan lebih mudah, karena jika kita ingin membedakan komposisi fungsi menggunakan limit definisi turunannya, kita harus melakukan banyak perhitungan.<\/p>\n<p> Di sisi lain, harus diingat bahwa aturan ini hanya digunakan untuk mencari turunan dari fungsi majemuk, dan bukan untuk jenis fungsi atau operasi apa pun dengan fungsi. Misalnya, kesalahan yang sangat umum adalah melakukan kesalahan dan menerapkan aturan rantai pada produk fungsional seperti berikut:<\/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-de22604d9af306981b71d39bd190df75_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\ln(x)\\cdot x^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"68\" style=\"vertical-align: -5px;\"><\/p>\n<p> \u274c<\/p>\n<p> Aturan rantai hanya dapat digunakan <strong>jika kita memiliki satu fungsi di dalam fungsi lainnya<\/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-ea93fb0bbc6f1ac5c2e26f2c5730627f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\ln(x^2)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"45\" style=\"vertical-align: -5px;\"><\/p>\n<p> \u2705 <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplos-de-derivadas-con-la-regla-de-la-cadena\"><\/span> Contoh turunan dengan aturan rantai<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Dengan mengetahui definisi aturan rantai, kita akan memperoleh beberapa fungsi dengan aturan rantai sebagai contoh. Ingatlah bahwa jika dalam contoh Anda tidak memahami bagaimana suatu fungsi diturunkan dengan aturan rantai, Anda dapat bertanya kepada kami di komentar!<\/p>\n<h3 class=\"wp-block-heading\"> Contoh 1<\/h3>\n<p> Dalam contoh ini, kita akan menggunakan aturan rantai untuk menurunkan logaritma natural dari x kuadrat:<\/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-056e8e809ecf361f98a9ab4a6509e1a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\ln(x^2)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"104\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Turunan logaritma natural sama dengan 1 kali argumennya, jadi turunannya<\/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-4537be7e40864f78dd4bf5a5cdfb53ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'\\bigl(g(x)\\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"62\" style=\"vertical-align: -7px;\"><\/p>\n<p> menjadi:<\/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-82f88b45158f1890a0e60b2496a1898e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\ln(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{1}{u}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"331\" style=\"vertical-align: -12px;\"><\/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-b66cdf74d9a89d4259495d799042e18c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f\\bigl(g(x)\\bigr)=\\ln(x^2) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'\\bigl(g(x)\\bigr)=\\cfrac{1}{x^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"396\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Sebaliknya, turunan x yang dipangkatkan dua adalah 2x:<\/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-0f0a7b2d096f09aacc349fe800f5ae6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"g(x)=x^2\\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} g'(x)=2x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"313\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Terakhir, kita menghitung turunan seluruh fungsi dengan menerapkan aturan rantai. Turunan dari fungsi komposit tersebut adalah hasil kali dua turunan yang baru kita temukan:<\/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-33416ac7184def2290a0a84cbd55a9b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"z(x)=f\\bigl(g(x)\\bigr) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black}z'(x)=f'\\bigl(g(x)\\bigr)\\cdot g'(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"450\" style=\"vertical-align: -7px;\"><\/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-6ad6b9a4a227664b616ffeef61781e54_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\ln(x^2) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{1}{x^2}\\cdot 2x = \\cfrac{2x}{x^2}=\\cfrac{2}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"458\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"> Contoh 2<\/h3>\n<p> Dalam contoh kedua ini, kita akan menurunkan fungsi potensial berdasarkan polinomial:<\/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-1fd64daeb147f4af91e5eb8518621081_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\left(3x^2+4x-5\\right)^3\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"179\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p> Untuk memperoleh suatu pangkat, kita perlu menempatkan eksponen asli di depannya dan mengurangkan satu satuan dari eksponen tersebut, sehingga turunan fungsi potensial tanpa menerapkan aturan rantai adalah:<\/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-9d0e7fc8a11cbd2103465a57128a9db4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f\\bigl(g(x)\\bigr)=\\left(3x^2+4x-5\\right)^3 \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'\\bigl(g(x)\\bigr)=3\\left(3x^2+4x-5\\right)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"52\" width=\"582\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p> Sekarang kita simpulkan apa yang ada di dalam tanda kurung:<\/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-dad3522b805cdb0c38e771bc6e630f50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"g(x)=3x^2+4x-5\\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} g'(x)=6x+4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"424\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Dan terakhir, kita menggunakan aturan rantai untuk menyelesaikan turunan seluruh fungsi, yang merupakan perkalian dua turunan yang dihitung sebelumnya: <\/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-33416ac7184def2290a0a84cbd55a9b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"z(x)=f\\bigl(g(x)\\bigr) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black}z'(x)=f'\\bigl(g(x)\\bigr)\\cdot g'(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"450\" style=\"vertical-align: -7px;\"><\/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-4d4d80ad509263a9791efd441621d183_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\left(3x^2+4x-5\\right)^3 \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=3\\left(3x^2+4x-5\\right)^2\\cdot (6x+4)\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"582\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"> Contoh 3<\/h3>\n<p> Dalam hal ini, kita akan mencari turunan sinus dari x pangkat tiga ditambah 7x:<\/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-42b994d7e38385bd61050cc50428beeb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{sen}(x^3+7x)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"154\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Memang benar, ini adalah komposisi fungsi karena kita mempunyai fungsi x <sup>3<\/sup> +7x di dalam fungsi sinus, oleh karena itu kita dapat menggunakan aturan rantai untuk mencari turunan dari fungsi komposit tersebut.<\/p>\n<p> Di satu sisi, turunan sinus adalah kosinus, sehingga turunan fungsi luarnya adalah kosinus dengan argumen yang sama dengan sinus:<\/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-e4b784343e9e483f8f3e2bb0cd465335_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f\\bigl(g(x)\\bigr)=\\text{sen}(x^3+7x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'\\bigl(g(x)\\bigr)=\\text{cos}(x^3+7x)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"522\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p> Sebaliknya turunan dari x <sup>3<\/sup> +7x adalah 3x <sup>2<\/sup> +7.<\/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-395ee5862baf231657c05660e22bbd42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"g(x)=x^3+7x\\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} g'(x)=3x^2+7\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"392\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Oleh karena itu, turunan fungsi komposit adalah hasil kali kedua turunannya: <\/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-33416ac7184def2290a0a84cbd55a9b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"z(x)=f\\bigl(g(x)\\bigr) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black}z'(x)=f'\\bigl(g(x)\\bigr)\\cdot g'(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"450\" style=\"vertical-align: -7px;\"><\/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-5b44098b23fd39005532d5f42593f585_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{sen}(x^3+7x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\text{cos}(x^3+7x)\\cdot (3x^2+7)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"555\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-derivadas-con-la-regla-de-la-cadena\"><\/span> Menyelesaikan latihan turunan dengan aturan rantai<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Latihan 1<\/h3>\n<p> Turunkan fungsi komposit berikut menggunakan 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-b84ea805fb8c56d493151d0f9b72b628_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\left(5x^2-6x\\right)^3\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"149\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Lihat solusinya<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Fungsi eksterior merupakan fungsi potensial, sehingga untuk menghitung turunannya harus menggunakan rumus berikut: <\/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-82e232ad4bd7b0f1b4b93625bd8dcf2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f\\bigl(g(x)\\bigr)=a\\bigl(g(x)\\bigr)^n \\ \\longrightarrow \\ f'\\bigl(g(x)\\bigr)=n\\cdot a\\bigl(g(x)\\bigr)^{n-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"397\" style=\"vertical-align: -7px;\"><\/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-058311927ac43c45c1de7d799d802310_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f\\bigl(g(x)\\bigr)=\\left(5x^2-6x\\right)^3\\ \\longrightarrow \\ f'\\bigl(g(x)\\bigr)= 3\\left(5x^2-6x\\right)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"415\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan kemudian kita menghitung turunan dari fungsi di dalamnya. Ini adalah pengurangan pangkat, jadi untuk menghitung turunannya Anda harus menerapkan rumus berikut untuk setiap sukunya: <\/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-eda0577ba91756ce6852219b0b1bf4c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=ax^n \\ \\longrightarrow \\ f'(x)=n\\cdot ax^{n-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"267\" 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-b4b1fc6c2a94bb4e9833be8140196f4b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"g(x)=5x^2-6x\\ \\longrightarrow\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"163\" style=\"vertical-align: -5px;\"><\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1c9d99adf81a8861bbd2dee3b8a7fcee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"g'(x)=2\\cdot 5x^1-1 \\cdot 6 x^0 =10x-6\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"262\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Singkatnya, turunan fungsi komposit adalah hasil kali dua turunan yang ditemukan: <\/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-8abee37ef3d49cc56596417a2e31618f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"z(x)=f\\bigl(g(x)\\bigr) \\ \\longrightarrow \\ z'(x)=f'\\bigl(g(x)\\bigr)\\cdot g'(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"342\" style=\"vertical-align: -7px;\"><\/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-f3e6ffbcb906ced150b00cf463b56434_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\left(5x^2-6x\\right)^3 \\ \\longrightarrow \\ \\bm{f'(x)= 3\\left(5x^2-6x\\right)^2\\cdot (10x-6)}\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"450\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Latihan 2<\/h3>\n<p> Selesaikan turunan fungsi komposit berikut menggunakan 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-c6ecb2411e5245a58c614748280b4568_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=-3\\left(5x^5+9x^3\\right)^4\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"182\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Lihat solusinya<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Pertama, kita cari turunan fungsi eksteriornya:<\/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-2e5d21c7196c7ebaeaf6ca11762ca251_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} f\\bigl(g(x)\\bigr) \\ \\longrightarrow \\ f'\\bigl(g(x)\\bigr) &amp; =4 \\cdot ( -3) \\left(5x^5+9x^3\\right)^3 \\\\[1.5ex]&amp;=-12\\left(5x^5+9x^3\\right)^3 \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"69\" width=\"357\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan sekarang kita selesaikan turunan dari fungsi interior:<\/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-fe839a2f2eb9412f63700dab70bf18f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"g(x)=5x^5+9x^3\\ \\longrightarrow \\ g'(x)=25x^4+27x^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"335\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu, turunan dari seluruh fungsi adalah: <\/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-8abee37ef3d49cc56596417a2e31618f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"z(x)=f\\bigl(g(x)\\bigr) \\ \\longrightarrow \\ z'(x)=f'\\bigl(g(x)\\bigr)\\cdot g'(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"342\" style=\"vertical-align: -7px;\"><\/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-0fe8c7e374a30ed8bcf0a83cea68d6bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=-3\\left(5x^5+9x^3\\right)^4 \\ \\longrightarrow \\ \\bm{f'(x)=-12\\left(5x^5+9x^3\\right)^3\\cdot \\left(25x^4+27x^2\\right)}\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"549\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Latihan 3<\/h3>\n<p> Hitung turunan komposisi fungsi berikut dengan 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-555bcc9c8b61b47c73e2014749954305_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=e^{2x^3}\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"87\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Lihat solusinya<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Ini adalah fungsi eksponensial, jadi untuk menghitung turunannya Anda harus menerapkan rumus berikut: <\/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-f52dc9e8ea936ea4de492bb3be18ebb6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=e^{x} \\ \\longrightarrow \\ f'(x)=e^{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"204\" 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-02e55bcf16e8288e1729ed5a4d06ed9e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f\\bigl(g(x)\\bigr)=e^{2x^3} \\ \\longrightarrow \\ f'\\bigl(g(x)\\bigr)= e^{2x^3}\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"281\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami juga membedakan fungsi dari eksponen fungsi:<\/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-b17b1c7b9b871d8404166d92d5cb0974_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"g(x)=2x^3 \\ \\longrightarrow \\ g'(x)=6x^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"221\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan kami menggunakan aturan rantai untuk mencari turunan dari fungsi komposit bilangan bulat: <\/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-8abee37ef3d49cc56596417a2e31618f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"z(x)=f\\bigl(g(x)\\bigr) \\ \\longrightarrow \\ z'(x)=f'\\bigl(g(x)\\bigr)\\cdot g'(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"342\" style=\"vertical-align: -7px;\"><\/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-ad4635f85ae781dd1565a8f6581d26c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=e^{2x^3} \\ \\longrightarrow \\ \\bm{f'(x)= e^{2x^3}\\cdot 6x^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"271\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Latihan 4<\/h3>\n<p> Temukan turunan dari fungsi komposit berikut menggunakan 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-a10feee3fd85abefa9ec5ea79c0cf223_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\sqrt[3]{\\text{sen}(x) +x }\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"158\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Lihat solusinya<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Ini adalah komposisi fungsi, karena kita memiliki fungsi sinusoidal dan fungsi linier dalam argumen fungsi irasional. Jadi kita hitung dulu turunan dari akarnya: <\/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-64a603462f094d4c699c56453463ca49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\sqrt[n]{x} \\ \\longrightarrow \\ f'(x)=\\cfrac{1}{n\\sqrt[n]{x^{n-1}}}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"265\" style=\"vertical-align: -16px;\"><\/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-e909efbe50930f94cce0b2485b060046_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f\\bigl(g(x)\\bigr)=\\sqrt[3]{\\text{sen}(x) +x } \\ \\longrightarrow \\ f'\\bigl(g(x)\\bigr)= \\cfrac{1}{3\\sqrt[3]{\\bigl(\\text{sen}(x) +x\\bigr)^2 }}\" title=\"Rendered by QuickLaTeX.com\" height=\"56\" width=\"455\" style=\"vertical-align: -30px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan sekarang kita mengambil argumen dari kaum radikal. Ini adalah penjumlahan fungsi, jadi turunannya adalah jumlah turunan tiap suku:<\/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-ffa1d177a8dfe81684225dffd555e6fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"g(x)=\\text{sen}(x) +x \\ \\longrightarrow \\ g'(x)=\\cos(x) + 1\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"326\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Jadi, turunan seluruh fungsi sama dengan perkalian dua turunan yang dihitung: <\/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-8abee37ef3d49cc56596417a2e31618f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"z(x)=f\\bigl(g(x)\\bigr) \\ \\longrightarrow \\ z'(x)=f'\\bigl(g(x)\\bigr)\\cdot g'(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"342\" style=\"vertical-align: -7px;\"><\/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-fad132b49a5faab86a3955efd5422973_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}f(x)=\\sqrt[3]{\\text{sen}(x)+x} \\ \\longrightarrow \\ f'(x)&amp; = \\cfrac{1}{3\\sqrt[3]{\\bigl(\\text{sen}(x) +x\\bigr)^2 }} \\cdot \\bigl(\\cos(x) + 1 \\bigr)\\\\[1.5ex]&amp;=\\cfrac{\\bm{\\cos(x) + 1}}{\\bm{3\\sqrt[3]{\\bigl(\\mathbf{sen}(x) +x\\bigr)^2} }}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"133\" width=\"509\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Latihan 5<\/h3>\n<p> Turunkan komposisi fungsi berikut menggunakan 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-09f37b7970003fc0221e15dccc157ccf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=3^{x^2+5}\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"100\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Lihat solusinya<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Untuk menerapkan aturan rantai, Anda harus mencari turunan pangkat dan polinomial, lalu mengalikannya. Jadi, kita memperoleh pangkat menggunakan rumus yang sesuai: <\/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-a9fb428e4d74e0f972130fde4e48ac0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=a^x \\ \\longrightarrow \\ f'(x)=a^x\\cdot \\ln (a)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"256\" 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-b1d18c3443d6398dcefba063ac556cbe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f\\bigl(g(x)\\bigr)=3^{x^2+5} \\ \\longrightarrow \\ f'\\bigl(g(x)\\bigr)= 3^{x^2+5}\\cdot  \\ln(3)\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"355\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kedua, kita memperoleh fungsi polinomial dari eksponen:<\/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-6070c83f8944ee39ae4e3e6e125bcc72_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"g(x)=x^2+5 \\ \\longrightarrow \\ g'(x)=2x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"235\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan aturan rantai memberitahu kita bahwa turunan seluruh fungsi adalah hasil kali turunan yang baru kita temukan: <\/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-8abee37ef3d49cc56596417a2e31618f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"z(x)=f\\bigl(g(x)\\bigr) \\ \\longrightarrow \\ z'(x)=f'\\bigl(g(x)\\bigr)\\cdot g'(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"342\" style=\"vertical-align: -7px;\"><\/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-bc0c78749a089e832984e3844345b6f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=3^{x^2+5} \\ \\longrightarrow \\ \\bm{f'(x)= 3^{x^2+5}\\cdot  \\ln(3) \\cdot 2x}\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"337\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Latihan 6 <\/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-6c85e9125bf4f54041c798dc4cc8975d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\ln \\bigl(4x^2 \\cdot \\cos(x) \\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"174\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Lihat solusinya<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Jelasnya, fungsi dalam soal ini adalah fungsi komposit, karena dalam argumen logaritma natural kita memiliki produk dari dua jenis fungsi yang berbeda. Jadi kita bedakan dulu logaritmanya: <\/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-a18d7f43ff1861389379485ae00db981_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\ln(x) \\ \\longrightarrow \\ f'(x)=\\cfrac{1}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"223\" style=\"vertical-align: -12px;\"><\/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-9b6ac8614a0671889738a762d0be9c29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f\\bigl(g(x)\\bigr)=\\ln \\bigl(4x^2 \\cdot \\cos(x) \\bigr) \\ \\longrightarrow \\ f'\\bigl(g(x)\\bigr)= \\cfrac{1}{4x^2 \\cdot \\cos(x) }\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"430\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kedua, kita memperoleh fungsi dari argumen logaritma. Ini adalah perkalian dua fungsi, jadi Anda harus menggunakan rumus berikut untuk melakukan turunannya: <\/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-cd35f94998c8450bd2e65e92eeecea2b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"z(x)=f(x) \\cdot g(x) \\ \\longrightarrow \\ z'(x)=f'(x)\\cdot g(x)+f(x) \\cdot g'(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"439\" 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-643ddf7ec82cbcc3bc685ceadf59da98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}g(x)=4x^2 \\cdot \\cos(x) \\ \\longrightarrow \\ g'(x) &amp; = 8x\\cdot \\cos(x) + 4x^2 \\cdot \\bigl(- \\text{sen}(x)\\bigr) \\\\[2ex] &amp; = 8x\\cdot \\cos(x) - 4x^2 \\cdot  \\text{sen}(x)\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"472\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Jadi, turunan seluruh fungsi, menurut aturan rantai, akan menjadi hasil kali kedua turunannya: <\/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-8abee37ef3d49cc56596417a2e31618f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"z(x)=f\\bigl(g(x)\\bigr) \\ \\longrightarrow \\ z'(x)=f'\\bigl(g(x)\\bigr)\\cdot g'(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"342\" style=\"vertical-align: -7px;\"><\/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-6912d0951fb85a61df21cbed282000f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}f'(x)&amp;= \\cfrac{1}{4x^2 \\cdot \\cos(x) } \\cdot \\bigl( 8x\\cdot \\cos(x) - 4x^2 \\cdot  \\text{sen}(x) \\bigr)\\\\[1.5ex]&amp;=\\cfrac{8x\\cdot \\cos(x) - 4x^2 \\cdot\\text{sen}(x)}{4x^2 \\cdot \\cos(x)}\\\\[1.5ex]&amp;=\\cfrac{\\bm{2\\cos(x) - x \\cdot }\\mathbf{sen}\\bm{(x)}}{\\bm{x \\cdot \\cos(x) }}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"169\" width=\"368\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Latihan 7<\/h3>\n<p> Selesaikan turunan fungsi berikut dengan menggunakan 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-3eb7a0c588b3aac39a2a4aa49a691598_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\log_9 (e^{x^2}-6x^7)\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"174\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Lihat solusinya<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Ini adalah komposisi fungsi, jadi kita akan membedakan logaritma dan argumennya secara terpisah lalu mengalikan turunannya.<\/p>\n<p class=\"has-text-align-left\"> Jadi, pertama-tama kita bedakan logaritmanya ke basis 9: <\/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-2c3339cb70e45253b4994a0c740202cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\log_a (x) \\ \\longrightarrow \\ f'(x)=\\cfrac{1}{x\\cdot \\ln (a)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"289\" 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-b0b4fc286244d6e5e35b8f7e94961314_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f\\bigl(g(x)\\bigr)=\\log_9 (e^{x^2}-6x^7) \\ \\longrightarrow \\ f'\\bigl(g(x)\\bigr)=\\cfrac{1}{\\bigl(e^{x^2}-6x^7\\bigr)\\cdot \\ln(9)}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"479\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan sekarang kita menghitung turunan dari argumen logaritma. Perhatikan bahwa bilangan e mempunyai fungsi dalam argumennya, yaitu fungsi komposit, jadi kita juga perlu menerapkan aturan rantai untuk menurunkan fungsi ini: <\/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-8abee37ef3d49cc56596417a2e31618f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"z(x)=f\\bigl(g(x)\\bigr) \\ \\longrightarrow \\ z'(x)=f'\\bigl(g(x)\\bigr)\\cdot g'(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"342\" style=\"vertical-align: -7px;\"><\/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-34065617ade6fb28fe66bc3f57a49cd9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"h(x)=e^{x^2} \\ \\longrightarrow \\ h'(x)=e^{x^2}\\cdot \\bigl(x^2\\bigr)' =e^{x^2}\\cdot 2x\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"348\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Jadi, turunan dari argumen bilangan bulat logaritma adalah:<\/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-1f7cd729a06f3cde16890b587693a667_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"g(x)= e^{x^2}-6x^7\\ \\longrightarrow \\ g'(x)=e^{x^2}\\cdot 2x - 42x^6\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"352\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan terakhir, turunan seluruh fungsi adalah hasil kali f'(g(x)) dan g'(x): <\/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-8abee37ef3d49cc56596417a2e31618f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"z(x)=f\\bigl(g(x)\\bigr) \\ \\longrightarrow \\ z'(x)=f'\\bigl(g(x)\\bigr)\\cdot g'(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"342\" style=\"vertical-align: -7px;\"><\/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-2a702df902c9f1eff66e14836a262c0a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}f'(x)&amp;=\\cfrac{1}{\\bigl(e^{x^2}-6x^7\\bigr)\\cdot \\ln(9)} \\cdot \\bigl(e^{x^2}\\cdot 2x - 42x^6\\bigr)\\\\[1.5ex]&amp;=\\cfrac{\\bm{e^{x^2}\\cdot 2x - 42x^6}}{\\bm{\\bigl(e^{x^2}-6x^7\\bigr)\\cdot \\ln(9)}}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"113\" width=\"342\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Latihan 8<\/h3>\n<p> Turunkan fungsi komposit berikut menggunakan 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-0fa2f9b67e41d5edc5bbef249f598359_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{sen}\\biggl( \\Bigl(9x^5 + \\cos(x) \\Bigr)^2 \\biggr)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"231\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Lihat solusinya<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Dalam latihan ini kita mempunyai komposisi beberapa fungsi, jadi kita harus menerapkan aturan rantai beberapa kali. Pertama-tama kita turunkan fungsi trigonometri dari sinus, yang turunannya adalah kosinus:<\/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-b6d04ed6e1b20f210641bb48c25c2c42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f\\bigl(g(x)\\bigr)=\\text{sen}\\biggl( \\Bigl(9x^5 + \\cos(x) \\Bigr)^2 \\biggr)\\ \\longrightarrow \\ f'\\bigl(g(x)\\bigr)=\\cos\\biggl( \\Bigl(9x^5 + \\cos(x) \\Bigr)^2 \\biggr)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"569\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan sekarang kita menghitung turunan dari argumen sinus menggunakan 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-8abee37ef3d49cc56596417a2e31618f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"z(x)=f\\bigl(g(x)\\bigr) \\ \\longrightarrow \\ z'(x)=f'\\bigl(g(x)\\bigr)\\cdot g'(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"342\" style=\"vertical-align: -7px;\"><\/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-2e1c2492990456e277e493c898cb3924_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} g(x)= \\Bigl( 9x^5 + \\cos(x) \\Bigr)^2 \\cdot g'(x) &amp;= 2\\Bigl(9x^5 + \\cos(x) \\Bigr) \\cdot \\Bigl(9x^5 + \\cos(x) \\Bigr)' \\\\[1.5ex]&amp;=2\\Bigl(9x^5 + \\cos(x) \\Bigr) \\cdot \\Bigl(45x^4-\\text{sen}(x)\\Bigr)\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"519\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Terakhir, kita memperoleh turunan seluruh komposisi fungsi dengan menerapkan kembali 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-8abee37ef3d49cc56596417a2e31618f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"z(x)=f\\bigl(g(x)\\bigr) \\ \\longrightarrow \\ z'(x)=f'\\bigl(g(x)\\bigr)\\cdot g'(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"342\" style=\"vertical-align: -7px;\"><\/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-db5ac3368ea7d37f280e0f538aaed1a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{f'(x)=\\cos } \\bm{\\biggl( \\Bigl(9x^5 + \\cos(x) \\Bigr)^2 \\biggr) \\cdot 2\\Bigl(9x^5 + \\cos(x) \\Bigr) \\cdot \\Bigl(45x^4-}\\mathbf{sen}\\bm{(x)\\Bigr)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"510\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"demostracion-de-la-regla-de-la-cadena\"><\/span> Bukti Aturan Rantai<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Terakhir, kita akan membuktikan rumus aturan rantai. Untuk melakukan ini, kita akan mulai dari definisi matematis dari turunan:<\/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-dc1699622d128f888c1f20599aeccf60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)=\\lim_{h \\to 0}\\frac{f(x+h)-f(x)}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"219\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> Misalkan <em>z<\/em> adalah fungsi yang terdiri dari dua fungsi:<\/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-6a650ba7c58d41f371d90a56e4d4fd4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"z=f\\bigl(g(x)\\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"90\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p> Maka turunan dari fungsi <em>z<\/em> yang menerapkan definisi tersebut adalah:<\/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-9419bc1d5617600c2ffea842822efed7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle z'(x)=\\lim_{h \\to 0}\\frac{f\\bigl(g(x+h)\\bigr)-f\\bigl(g(x)\\bigr)}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"269\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> Seperti yang sudah Anda ketahui, Anda bisa mengalikan dan membagi pecahan dengan suku yang sama, karena hal ini tidak mengubah hasilnya. Oleh karena itu, kita dapat melanjutkan ke langkah berikutnya:<\/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-18ff4ca3bcd8ba04a25aa0187b3b5b3a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle z'(x)=\\lim_{h \\to 0}\\frac{f\\bigl(g(x+h)\\bigr)-f\\bigl(g(x)\\bigr)}{h}\\cdot \\frac{g(x+h)-g(x)}{g(x+h)-g(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"405\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Kami mengatur ulang penyebut 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-9059bbff916941c2e161b6d127ec654e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle z'(x)=\\lim_{h \\to 0}\\frac{f\\bigl(g(x+h)\\bigr)-f\\bigl(g(x)\\bigr)}{g(x+h)-g(x)}\\cdot \\frac{g(x+h)-g(x)}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"405\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Dengan menerapkan sifat-sifat limit, kita dapat membagi limit di atas menjadi dua. Karena limit suatu hasil kali sama dengan hasil kali limitnya:<\/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-220a40fee3825089394f3d6e5578c4eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle z'(x)=\\lim_{h \\to 0}\\frac{f\\bigl(g(x+h)\\bigr)-f\\bigl(g(x)\\bigr)}{g(x+h)-g(x)}\\cdot \\lim_{h \\to 0}\\frac{g(x+h)-g(x)}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"436\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Dan ungkapan ini setara dengan berikut ini:<\/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-8188c6fac1c61928975e7a8c02ac79c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle z'(x)=f'\\bigl(g(x)\\bigr)\\cdot g'(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"175\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p> Oleh karena itu, rumus aturan rantai terbukti karena kita mendapatkannya dari definisi turunan.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Di sini Anda akan mempelajari apa itu aturan rantai dan cara menurunkan fungsi menggunakan aturan rantai. Selain itu, Anda akan dapat melihat beberapa contoh turunan yang diselesaikan dengan aturan rantai dan Anda bahkan dapat berlatih dengan latihan penyelesaian langkah demi langkah pada turunan yang menerapkan aturan rantai. Apa aturan rantainya? Aturan rantai adalah rumus yang &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/id\/rantai-aturan-turunan\/\"> <span class=\"screen-reader-text\">Aturan rantai (turunan)<\/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-37","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>\u25b7 Aturan rantai (turunan): latihan yang diselesaikan<\/title>\n<meta name=\"description\" content=\"Kami menjelaskan cara menurunkan fungsi komposit dengan aturan rantai. Dengan menyelesaikan latihan turunan dengan menerapkan aturan rantai.\" \/>\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\/rantai-aturan-turunan\/\" \/>\n<meta property=\"og:locale\" content=\"id_ID\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u25b7 Aturan rantai (turunan): latihan yang diselesaikan\" \/>\n<meta property=\"og:description\" content=\"Kami menjelaskan cara menurunkan fungsi komposit dengan aturan rantai. 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