{"id":30,"date":"2023-09-17T11:02:59","date_gmt":"2023-09-17T11:02:59","guid":{"rendered":"https:\/\/mathority.org\/id\/berasal-dari-kosinus\/"},"modified":"2023-09-17T11:02:59","modified_gmt":"2023-09-17T11:02:59","slug":"berasal-dari-kosinus","status":"publish","type":"post","link":"https:\/\/mathority.org\/id\/berasal-dari-kosinus\/","title":{"rendered":"Turunan kosinus"},"content":{"rendered":"<p>Di sini Anda akan menemukan cara menurunkan fungsi kosinus (rumus). Anda akan dapat melihat contoh turunan fungsi kosinus dan berlatih dengan latihan langkah demi langkah. Selain itu, kami tunjukkan bukti rumusnya, berapakah turunan kedua dari cosinus dan bahkan turunan dari invers cosinus. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfcual-es-la-derivada-del-coseno\"><\/span> Berapakah turunan dari cosinus?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>Turunan fungsi kosinus adalah fungsi sinus termodifikasi tanda. Dengan kata lain, turunan cosinus x sama dengan dikurangi sinus x.<\/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-72551067d650b8d3797bc37497ec609d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{cos}(x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\text{sen}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"389\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Jika terdapat suatu fungsi dalam argumen kosinus, maka turunan kosinusnya adalah hasil kali dikurangi sinus fungsi tersebut dikalikan dengan turunan 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-ccc4f6fce30c027f8782a296a44b84b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{cos}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\text{sen}(u)\\cdot u'\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"416\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Rumus kedua setara dengan rumus pertama namun menerapkan aturan rantai. Jadi, secara ringkas rumus turunan cosinusnya adalah sebagai berikut: <\/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-du-cosinus.webp\" alt=\"turunan kosinus\" class=\"wp-image-1902\" width=\"428\" height=\"292\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplos-de-la-derivada-del-coseno\"><\/span> Contoh turunan kosinus<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Setelah kita mengetahui apa itu rumus kosinus, kami akan menjelaskan beberapa contoh turunan trigonometri jenis ini agar Anda tidak ragu lagi bagaimana cara menurunkan fungsi kosinus. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-1-derivada-del-coseno-de-2x\"><\/span> Contoh 1: Turunan dari kosinus 2x<span class=\"ez-toc-section-end\"><\/span><\/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-87c696135df266b2d8498b353bf03c36_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{cos}(2x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"114\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Dalam argumen kosinus kita tidak memiliki satu x pun, melainkan fungsi yang lebih kompleks. Oleh karena itu, kita perlu menggunakan rumus berikut untuk menurunkan 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-ccc4f6fce30c027f8782a296a44b84b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{cos}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\text{sen}(u)\\cdot u'\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"416\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Karena turunan 2x adalah 2, maka turunan cosinus 2x dikurangi sinus 2x dikalikan 2. <\/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-ba75c906f1694fe3fbd16fa61e0d288e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{cos}(2x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\text{sen}(2x)\\cdot 2=-2\\text{sen}(2x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"532\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-2-derivada-del-coseno-de-x-al-cuadrado\"><\/span> Contoh 2: Turunan dari kosinus x kuadrat<span class=\"ez-toc-section-end\"><\/span><\/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-89f1a1fc3f2d5e95aafbd2a37282f88c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{cos}(x^2)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"113\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Seperti pada contoh sebelumnya, dalam argumen cosinus kita mempunyai fungsi yang berbeda dari x, jadi kita akan menggunakan aturan rantai untuk menurunkan 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-ccc4f6fce30c027f8782a296a44b84b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{cos}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\text{sen}(u)\\cdot u'\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"416\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Maka turunan dari x <sup>2<\/sup> adalah 2x, maka turunan cosinus x yang dipangkatkan 2 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-a410f3316194c86b97a987b0ec7e9e6a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{cos}(x^2) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\text{sen}(x^2)\\cdot 2x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"437\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-3-derivada-del-coseno-al-cubo\"><\/span> Contoh 3: Turunan dari kosinus pangkat tiga<span class=\"ez-toc-section-end\"><\/span><\/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-dfa2e76d23ef3aeb2ab3ff8e20e2aa07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{cos}^3(2x^6-5x^3)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"178\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Fungsi kosinus dalam contoh ini terdiri dari fungsi lain, jadi kita perlu menerapkan rumus berikut untuk menyelesaikan 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-ccc4f6fce30c027f8782a296a44b84b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{cos}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\text{sen}(u)\\cdot u'\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"416\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Jadi, dengan menerapkan rumus tersebut, kita mendapatkan turunan dari 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-73284bcfb1d5647b2304e323e7fbaedf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}f'(x)&amp;=3\\text{cos}^2(2x^6-5x^3)\\cdot \\bigl(-\\text{sen}(2x^6-5x^3)\\bigr)\\cdot (12x^5-15x^2)\\\\[2ex]&amp;=-3\\text{cos}^2(2x^6-5x^3)\\cdot \\text{sen}(2x^6-5x^3)\\cdot (12x^5-15x^2)\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"467\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <u style=\"text-decoration-color:#ff951b;\">Untuk membedakan fungsi ini, Anda juga harus menggunakan rumus turunan<\/u> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/id\/turunan-dari-fungsi-potensial-daya\/\">fungsi potensial<\/a><\/span> . <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"segunda-derivada-del-coseno\"><\/span> Turunan kedua dari kosinus<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Selanjutnya kita akan melihat bahwa turunan kedua sinus dapat dengan mudah dihitung berkat karakteristik fungsi trigonometri.<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <u style=\"text-decoration-color:#ff951b;\"><strong>Catatan:<\/strong> Untuk memahami hal berikut ini, Anda perlu mengetahui<\/u> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/id\/turunan-sinus\/\">apa itu turunan sinus<\/a><\/span> .<\/p>\n<p> <strong>Turunan kedua kosinus x dikurangi kosinus x.<\/strong> Ini mungkin tampak aneh, tapi secara matematis memang seperti itu. Memang, turunan dari sinus adalah kosinus dan, oleh karena itu, dengan mendiferensiasikan kosinus x dua kali, kosinus diperoleh kembali tetapi dengan tanda yang dimodifikasi.<\/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-d0d9dda8a4031c367120b1f950da4391_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}f(x)=\\text{cos}(x)\\\\[1.5ex] \\quad\\color{orange}\\bm{\\downarrow}\\quad\\color{black} \\\\[1.5ex] f'(x)=-\\text{sen}(x)\\\\[2ex] \\quad\\color{orange}\\bm{\\downarrow}\\quad\\color{black} \\\\[1.5ex] f''(x)=-\\text{cos}(x)\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"157\" width=\"132\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Properti ini berubah jika argumen kosinusnya bukan x, karena dalam kasus ini kita menyeret suku 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-83d86bd6508f06b0723153b3b9254c1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}f(x)=\\text{cos}(u)\\\\[1.5ex] \\quad\\color{orange}\\bm{\\downarrow}\\quad\\color{black} \\\\[1.5ex] f'(x)=-\\text{sen}(u)\\cdot u' \\\\[1.5ex] \\quad\\color{orange}\\bm{\\downarrow}\\quad\\color{black} \\\\[1.5ex] f''(x)=-\\text{cos}(u)\\cdot u'^2 -\\text{sen}(u)\\cdot u'' \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"153\" width=\"263\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"derivada-del-coseno-inverso\"><\/span> Turunan dari kosinus terbalik<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Semua fungsi trigonometri mempunyai fungsi invers sehingga fungsi kosinus juga dapat diinvers. Demikian pula, kosinus terbalik dapat terdiferensiasi.<\/p>\n<p> <strong>Turunan invers cosinus<\/strong> suatu fungsi adalah dikurangi turunan fungsi tersebut dibagi akar kuadrat 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-307f91156ee9c404e9c1a1c0de56b102_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{cos}^{-1}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\cfrac{u'}{\\sqrt{1-u^2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"425\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> Ingatlah bahwa kosinus terbalik juga disebut arccosine.<\/p>\n<p> Misalnya turunan dari invers cosinus 3x 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-7ce426dcd95d21e43b182ef593520c16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{cos}^{-1}(3x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\cfrac{3}{\\sqrt{1-(3x)^2}}=-\\cfrac{3}{\\sqrt{1-9x^2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"571\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-la-derivada-del-coseno\"><\/span> Latihan soal turunan kosinus<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Hitung turunan fungsi kosinus 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-c1caebbb3b9acfa8cd25721299f9a22e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A) } f(x)=\\text{cos}(4x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"140\" 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-a89d6e415addae7423aa75362416686b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B) } f(x)=\\text{cos}(2x^3-5x+1)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"218\" 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-dad34285ad7ac07f34ef408c65cbb96c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C) } \\displaystyle f(x)=9\\text{cos}\\left(\\frac{x}{3}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"152\" 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-752bff297c20cdf69d6fcb45290be935_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D) } f(x)=\\text{cos}^5(x^2+3x)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"187\" 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-939dee8bcafba3f55812c5a13f27a309_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{E) } f(x)=\\text{cos}\\left(e^{5x}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"148\" 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-f6c7c3c5786d010b99f4e65b692dfe1d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{F) } \\displaystyle f(x)=9\\text{cos}\\left(\\frac{e^x}{5x}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"164\" 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-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8a3e1b1b2fe486d1c432a075c0028b62_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A) } f'(x)=-\\text{sen}(4x)\\cdot 4 =-4\\text{sen}(4x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"285\" 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-8bc2f2e1676bb5f0d4ca231bd35b2b12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B) } f'(x)=-\\text{sen}(2x^3-5x+1)\\cdot (6x^2-5)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"321\" 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-cc5cf86d30b34d4cd1a794a4d2ee6a5e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C) } \\displaystyle f'(x)=-9\\text{sen}\\left(\\frac{x}{3}\\right)\\cdot \\frac{1}{3} =-3\\text{sen}\\left(\\frac{x}{3}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"306\" 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-f168f2e897b18c662f567a25ff09e881_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D) } f'(x)=-5\\text{cos}^4(x^2+3x)\\cdot \\text{sen}(x^2+3x)\\cdot (2x+3)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"401\" 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-fe49736d7a1ce1736679e8c25bc4a66b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{E) } f'(x)=-\\text{sen}\\left(e^{5x}\\right)\\cdot e^{5x}\\cdot 5=-5\\text{sen}\\left(e^{5x}\\right)\\cdot e^{5x}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"377\" 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-9f4645fb77435daec6f696cffbd54884_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}\\text{F) }\\displaystyle f'(x)&amp;=-9\\text{sen}\\left(\\frac{e^x}{5x}\\right)\\cdot \\frac{e^x\\cdot 5x-e^x\\cdot 5}{(5x)^2}\\\\[2ex]&amp;=-9\\text{sen}\\left(\\frac{e^x}{5x}\\right)\\cdot \\frac{5e^x(x-1)}{25x^2}\\\\[2ex]&amp;=-9\\text{sen}\\left(\\frac{e^x}{5x}\\right)\\cdot \\frac{e^x(x-1)}{5x^2}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"172\" width=\"310\" style=\"vertical-align: 0px;\"><\/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-derivada-del-coseno\"><\/span> Bukti turunan kosinus<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Terakhir, kita akan mendemonstrasikan secara matematis rumus turunan kosinus x. Untuk melakukan ini, kita akan menggunakan definisi turunan, yang sesuai dengan limit 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-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> Kita akan buktikan cosinusnya, jadi fungsinya adalah cos(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-a00c11698e4b4f5caf0f227e18be8656_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)=\\lim_{h \\to 0}\\frac{\\text{cos}(x+h)-\\text{cos}(x)}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"245\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> Kita tidak dapat menyelesaikan batasan ini dengan substitusi, karena kita akan berakhir pada ketidakpastian. Namun, kita dapat menyatakan kosinus suatu jumlah dengan cara lain dengan menerapkan identitas trigonometri 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-5e06f1728cce31fb5650ba149b8e5b9a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{cos}(a+b)=\\text{cos}(a)\\text{cos}(b)-\\text{sen}(a)\\text{sen}(b)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"307\" 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-db64449e24b11a613417ebce4c7c7a85_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)=\\lim_{h \\to 0}\\frac{\\text{cos}(x)\\text{cos}(h)-\\text{sen}(x)\\text{sen}(h)-\\text{cos}(x)}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"380\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> Langkah selanjutnya adalah memisahkan pecahan menjadi dua pecahan dan mengambil faktor persekutuan kosinusnya:<\/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-c7c1cd89cf290b01d7d72fc8084f6529_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)=\\lim_{h \\to 0}\\left[\\frac{\\text{cos}(x)\\bigl(\\text{cos}(h)-1\\bigr)}{h}-\\frac{\\text{sen}(x)\\text{sen}(h)}{h}\\right]\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"380\" style=\"vertical-align: -23px;\"><\/p>\n<\/p>\n<p> Limit suatu pengurangan sama dengan pengurangan limitnya, maka:<\/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-739fc9a2280c7da1bf2ea830ee5ec88c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)=\\lim_{h \\to 0}\\frac{\\text{cos}(x)\\bigl(\\text{cos}(h)-1\\bigr)}{h}-\\lim_{h \\to 0}\\frac{\\text{sen}(x)\\text{sen}(h)}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"393\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> Kosinus dari x dan sinus dari x tidak bergantung pada h, sehingga kita dapat mengeluarkannya ke luar batas:<\/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-bfbc83e5a84d91a0f6d98418a4f0041c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)=\\text{cos}(x)\\lim_{h \\to 0}\\frac{\\text{cos}(h)-1}{h}-\\text{sen}(x)\\lim_{h \\to 0}\\frac{\\text{sen}(h)}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"383\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> Dengan menggunakan perhitungan limit dengan ekuivalen yang sangat kecil, kita menyimpulkan bahwa limit pertama adalah 0 dan limit kedua adalah 1. 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-f0c2ed1188b80356d05d6188fab5ca47_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)=\\text{cos}(x)\\cdot 0-\\text{sen}(x)\\cdot 1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"223\" 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-9f33ae6c9b18e01ba654772f22cab6d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)=-\\text{sen}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"124\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Dan kita sudah sampai pada rumus turunan fungsi kosinus, sehingga persamaannya terbukti.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Di sini Anda akan menemukan cara menurunkan fungsi kosinus (rumus). Anda akan dapat melihat contoh turunan fungsi kosinus dan berlatih dengan latihan langkah demi langkah. Selain itu, kami tunjukkan bukti rumusnya, berapakah turunan kedua dari cosinus dan bahkan turunan dari invers cosinus. Berapakah turunan dari cosinus? Turunan fungsi kosinus adalah fungsi sinus termodifikasi tanda. Dengan &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/id\/berasal-dari-kosinus\/\"> <span class=\"screen-reader-text\">Turunan kosinus<\/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-30","post","type-post","status-publish","format-standard","hentry","category-derivatif"],"yoast_head":"<!-- This site is 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