{"id":39,"date":"2023-09-17T10:58:59","date_gmt":"2023-09-17T10:58:59","guid":{"rendered":"https:\/\/mathority.org\/id\/persamaan-tangen\/"},"modified":"2023-09-17T10:58:59","modified_gmt":"2023-09-17T10:58:59","slug":"persamaan-tangen","status":"publish","type":"post","link":"https:\/\/mathority.org\/id\/persamaan-tangen\/","title":{"rendered":"Persamaan garis singgung"},"content":{"rendered":"<p>Pada artikel ini kita akan melihat <strong>cara mencari persamaan garis singgung<\/strong> suatu kurva. Selain itu, Anda dapat berlatih dengan latihan yang diselesaikan dengan tingkat kesulitan berbeda. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ecuacion-de-la-recta-tangente-a-una-funcion-en-un-punto\"><\/span> Persamaan garis singgung suatu fungsi di suatu titik <span class=\"ez-toc-section-end\"><\/span><\/h2>\n<div style=\"padding-top: 23px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 30px; border: 2px dashed #FF9B28; border-radius:20px;\">\n<p style=\"text-align:left\"> <strong>Persamaan garis singgung<\/strong> fungsi f(x) di titik x=x <sub>0<\/sub> 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-326424811181144df35c0b94ce50c462_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y -y_0= m(x-x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p align=\"LEFT\"> Dimana titik P(x <sub>0<\/sub> ,y <sub>0<\/sub> ) merupakan titik dimana garis singgung dan fungsinya berimpit. Dan kemiringan garis singgung, m, sama dengan turunan kurva di titik x <sub>0<\/sub> , yaitu m=f'(x <sub>0<\/sub> ). <\/p>\n<\/div>\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\/equation-de-la-tangente-ligne.webp\" alt=\"persamaan tangen\" class=\"wp-image-2306\" width=\"463\" height=\"461\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Pada gambar di atas Anda dapat melihat sebuah kurva<\/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-a7ee323bc5a3f73ad5e066b13bed5504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"34\" style=\"vertical-align: -5px;\"><\/p>\n<p> diwakili dengan warna biru dan garis oranye yang bersinggungan dengan fungsi tersebut<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a7ee323bc5a3f73ad5e066b13bed5504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"34\" style=\"vertical-align: -5px;\"><\/p>\n<p> Tentang<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bd5304ac1643ba3660a7efa36ade1983_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=x_0\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"51\" style=\"vertical-align: -3px;\"><\/p>\n<p> , karena mereka hanya memiliki kesamaan dalam hal ini. Nah, persamaan garis singgungnya adalah<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-326424811181144df35c0b94ce50c462_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y -y_0= m(x-x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -5px;\"><\/p>\n<p> , dan kemiringannya adalah<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5c606eb4b268b71562672c32a0461053_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=f'(x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"85\" style=\"vertical-align: -5px;\"><\/p>\n<p> . <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"como-hallar-la-ecuacion-de-la-recta-tangente\"><\/span> Cara mencari persamaan tangen<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Untuk mencari persamaan garis singgung suatu fungsi di suatu titik, Anda perlu melakukan:<\/p>\n<ol style=\"color:#FF8A05; font-weight: bold;\">\n<li style=\"margin-bottom:8px\"> <span style=\"color:#101010;font-weight: normal;\">Mencari kemiringan garis singgung dengan menghitung turunan fungsi di titik singgung tersebut.<\/span><\/li>\n<li style=\"margin-bottom:8px\"> <span style=\"color:#101010;font-weight: normal;\">Tentukan sebuah titik pada garis singgung tersebut.<\/span><\/li>\n<li> <span style=\"color:#101010;font-weight: normal;\"><strong>Temukan persamaan garis singgung<\/strong> menggunakan perhitungan kemiringan dan titik garis singgung.<\/span> <\/li>\n<\/ol>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-de-la-ecuacion-de-la-recta-tangente-a-una-curva\"><\/span> Contoh persamaan garis singgung suatu kurva<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Setelah kita melihat teori persamaan tangen, mari kita lihat cara menghitung persamaan tangen dengan menyelesaikan contoh langkah demi langkah:<\/p>\n<ul>\n<li> Hitung persamaan garis singgung kurva\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ea8efcaacce7a90d3cc483105986c47c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2+x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<p> Tentang<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3330a01aa4d7d81947b71297d8623d3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=1\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\"><\/p>\n<p> .<\/li>\n<\/ul>\n<p> Kita tahu bahwa persamaan tangen selalu berbentuk 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-326424811181144df35c0b94ce50c462_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y -y_0= m(x-x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Hal pertama yang harus dilakukan adalah menghitung kemiringan garis. Jadi, kemiringan garis singgung,<\/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-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p> , akan menjadi nilai turunan kurva pada titik singgung x=1, yaitu<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a69005ee8bf2d80d73b989ad0cedccd7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=f'(1).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"81\" style=\"vertical-align: -5px;\"><\/p>\n<p> Maka dari itu kita bedakan fungsinya lalu kita hitung <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-eb68a498d3bd60e51d3dc230691f886c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(1):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" 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-92c9a5ee4068789701733f793fbac622_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2+x \\quad \\longrightarrow \\quad f'(x)=2x+1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"293\" 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-d96270e9b7d7c3cae8baea602cea53bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(1)= 2\\cdot 1+1=2+1=3\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" 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-2443a93bff265c6b8ba692ef8d14f633_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=f'(1)=3\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"110\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Setelah kita mengetahui nilai dari<\/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-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p> , kita perlu menemukan suatu titik<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8e531a80c13865d1ad612bd3f634efa2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x_0,y_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"54\" style=\"vertical-align: -5px;\"><\/p>\n<p> garis singgung untuk melengkapi persamaan garis singgung.<\/p>\n<p> <strong>Persamaan garis singgung dan kurva selalu mempunyai titik persekutuan<\/strong> , yang dalam hal ini 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-3330a01aa4d7d81947b71297d8623d3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=1\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\"><\/p>\n<p> . Oleh karena itu, seperti kurva<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a7ee323bc5a3f73ad5e066b13bed5504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"34\" style=\"vertical-align: -5px;\"><\/p>\n<p> melewati titik ini, kita dapat mencari komponen titik lainnya dengan menghitung <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-32045357853caad8774629c95963835d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(1):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" 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-ea8efcaacce7a90d3cc483105986c47c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2+x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"108\" 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-7b2601f20b100f2635bc0342175b4627_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(1)=1^2+1=2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"136\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Oleh karena itu, titik singgungnya 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-d26257abb9047188ab3e3887f447e20a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(1,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"52\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Baik kurva maupun garis singgung melewati titik ini, jadi kita juga dapat menggunakannya untuk mencari persamaan garis singgung.<\/p>\n<p> Yang tersisa hanyalah mengganti nilai kemiringan dan titik singgung yang ditemukan ke dalam persamaannya:<\/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-0321e19825c08a1f47a00b2cf625088f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{c} y -y_0= m(x-x_0) \\\\[3ex] m=3 \\qquad P(1,2) \\end{array} \\right\\} \\longrightarrow \\ y -2= 3(x-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"344\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Singkatnya, persamaan tangennya 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-037e5c42a4adef3e5ba970a66b8d3459_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y-2=3(x-1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"126\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<hr class=\"wp-block-separator has-text-color has-background is-style-wide\" style=\"background-color:#1976d2;color:#1976d2\">\n<p> Anda juga dapat menyatakan persamaan garis singgung dengan persamaan garis eksplisit: <\/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-59097f2ef899c7c608e2527467021b23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y=3x-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"82\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<hr class=\"wp-block-separator has-text-color has-background is-style-wide\" style=\"background-color:#1976d2;color:#1976d2\">\n<p> Di bawah ini Anda dapat melihat kurva yang diwakili<\/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-ea8efcaacce7a90d3cc483105986c47c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2+x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<p> dan garis singgungnya <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e2883f0b53c531552fde7ff189f83165_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=1,\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"47\" style=\"vertical-align: -4px;\"><\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1020651b62576571e0ac9c0cb65dd287_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-2=3(x-1):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"136\" style=\"vertical-align: -5px;\"><\/p>\n<\/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\/equation-de-la-tangente-a-une-courbe-en-un-point.webp\" alt=\"persamaan garis singgung kurva di suatu titik\" class=\"wp-image-2318\" width=\"445\" height=\"434\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Seperti yang Anda lihat, kurvanya<\/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-ea8efcaacce7a90d3cc483105986c47c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2+x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<p> dan garis singgung<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3cb3f64ce416f3a5c6cc80c11cae9afb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-2=3(x-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"126\" style=\"vertical-align: -5px;\"><\/p>\n<p> mereka hanya mempunyai kesamaan maksudnya<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-42a32282c97c3b8d9f90b2f1418844d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(1,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<p> , persis seperti yang kami hitung. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-la-ecuacion-de-la-recta-tangente\"><\/span> Latihan soal persamaan tangen diselesaikan<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Latihan 1<\/h3>\n<p> Hitung persamaan garis singgung kurva<\/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-99f623c4fede3b664682c5cbc1aab81d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=2x^2-4x+3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"156\" style=\"vertical-align: -5px;\"><\/p>\n<p> Tentang <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-71e6033606cd14039ab202fb7a18c50e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=2 .\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\"><\/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\"> Persamaan tangen akan selalu berbentuk 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-3441e1da8c7da5805b1133af77b14f60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-y_0=m(x-x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <strong>Langkah 1: Hitung kemiringan garis singgung<\/strong><\/p>\n<p class=\"has-text-align-left\"> Kemiringan, <em>m<\/em> , adalah nilai turunan kurva di titik singgung. Oleh karena itu, dalam hal 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-c7192bf7bd4300d7d77fe084134d6849_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m = f'(2):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"86\" 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-ab9e97205d5c9f2fbfcb085cbfdbdd75_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=2x^2-4x+3 \\ \\longrightarrow \\ f'(x)= 4x-4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"319\" 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-e3cb482f7e68631c8dcc5705ac1257d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(2)= 4\\cdot 2-4=8-4=4\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" 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-b0e7f5cd5a44a120769c9d3a1eae02c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=f'(2)=4\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"110\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <strong>Langkah 2: Temukan titik pada garis singgung<\/strong><\/p>\n<p class=\"has-text-align-left\"> Persamaan garis singgung dan kurva selalu mempunyai titik persekutuan, yang dalam hal ini 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-c657687cbbf5ea9a7545edb42190e592_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\"><\/p>\n<p> . Oleh karena itu, seperti kurva<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a7ee323bc5a3f73ad5e066b13bed5504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"34\" style=\"vertical-align: -5px;\"><\/p>\n<p> melewati titik ini, kita dapat mencari komponen titik lainnya dengan menghitung <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f026e401162db03299777455b748b308_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(2):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" 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-99f623c4fede3b664682c5cbc1aab81d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=2x^2-4x+3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"156\" 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-67ff5b86b2842fefbdf2ddc7c2df39f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(2)=2\\cdot 2^2-4\\cdot 2+3 =2 \\cdot 4 -8 +3 = 3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"326\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Jadi, titik yang dilalui kurva dan garis singgung adalah titik<\/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-d744fa34f41ed1bbe3fdf2c5ad7f55a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(2,3).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"43\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <strong>Langkah 3: Tulis persamaan tangennya<\/strong><\/p>\n<p class=\"has-text-align-left\"> Yang tersisa hanyalah mengganti nilai kemiringan dan titik singgung yang ditemukan ke dalam persamaannya:<\/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-1622c6ecd4d43bb4fc4901b437464652_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{c} y -y_0= m(x-x_0) \\\\[3ex] m=4 \\qquad P(2,3) \\end{array} \\right\\} \\longrightarrow \\ y -3= 4(x-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"344\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu, persamaan tangennya 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-c1233301c390a75095fc24bd8765e081_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y -3= 4(x-2)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"126\" 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 2<\/h3>\n<p> Hitung persamaan garis singgung kurva<\/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-d1309dcf6b647174b562cb71ab600c1c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)=-3x^2+2x\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"139\" style=\"vertical-align: -5px;\"><\/p>\n<p> di titik asal koordinat. <\/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\"> Asal koordinat mengacu pada titik<\/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-d5f3123d35179a39bd727675fca259c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(0,0).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"43\" style=\"vertical-align: -5px;\"><\/p>\n<p> Oleh karena itu kita harus menghitung garis singgung fungsi tersebut di titik tersebut<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-791f3561f68c75b943d5af446c9f988f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(0,0) .\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"43\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Pertama, kita menentukan nilai kemiringan garis singgung dengan menghitung turunan di titik asal koordinat: <\/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-e3f5e3bd3a06eb5e2831d90e5fc0f31d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=-3x^2+2x \\ \\longrightarrow \\  f'(x)= -6x+2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"315\" 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-d07228d62a796c695cb75841830d0e17_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(0)= -6\\cdot 0+2=2\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"168\" 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-075eebc763002b84e54211e61242356f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=f'(0)=2\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"109\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dalam hal ini, kita sudah mengetahui titik yang dilalui garis singgung. Karena pernyataan tersebut menyatakan bahwa garis tersebut harus bersinggungan dengan kurva di titik asal, yaitu di titik<\/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-d5f3123d35179a39bd727675fca259c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(0,0).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"43\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Jadi titik dimana kurva dan garis singgungnya adalah titik<\/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-d5f3123d35179a39bd727675fca259c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(0,0).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"43\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Terakhir, substitusikan nilai kemiringan dan titik singgung yang ditemukan ke dalam persamaan 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-de8e4e9dbb7a5bca1d591612abcf7730_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{c} y -y_0= m(x-x_0) \\\\[3ex] m=2 \\qquad P(0,0) \\end{array} \\right\\} \\longrightarrow \\ y -0= 2(x-0)\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"344\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kesimpulannya, persamaan tangennya 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-19b9a613ed0c41d6c98ab37c6a0a1331_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y -0= 2(x-0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"126\" 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-329a7fc0d44a0b32cbb521e81ee50db6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y = 2x}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"52\" style=\"vertical-align: -4px;\"><\/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 garis singgung kurva 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-33130a168a4e20b536fb742b8ce2a662_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2-2x-1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"146\" style=\"vertical-align: -5px;\"><\/p>\n<p> yang sejajar dengan kanan<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3b4e35094bc85458a54e2b47228f9c39_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-4x-6=0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"113\" style=\"vertical-align: -4px;\"><\/p>\n<p> . <\/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 soal ini kita diberitahu bahwa garis singgung harus sejajar dengan garis<\/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-02c5cab1d3747c5baa1ded66f3055f61_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-4x-6=0 .\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"117\" style=\"vertical-align: -4px;\"><\/p>\n<p> Dan dua garis dikatakan sejajar jika mempunyai kemiringan yang sama. Oleh karena itu, garis singgung harus mempunyai kemiringan yang sama dengan garis<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3dee6df2aaaef2e062c41a79057de62e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-4x-6=0.\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"117\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Artinya kita perlu mencari kemiringan garis<\/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-02c5cab1d3747c5baa1ded66f3055f61_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-4x-6=0 .\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"117\" style=\"vertical-align: -4px;\"><\/p>\n<p> Untuk melakukan ini, kami menghapus variabel dan:<\/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-6390ce01aebdda3a7305c4dd1e55d4aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-4x-6=0 \\ \\longrightarrow \\ y =4x+6\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"246\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Jadi kemiringan garisnya<\/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-824b16de72dde879834460d93bd88610_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=4x+5\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"82\" style=\"vertical-align: -4px;\"><\/p>\n<p> adalah 4, karena kemiringan suatu garis adalah bilangan yang mengalikan x jika ynya jelas.<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu kemiringan garis singgungnya juga harus 4, karena agar sejajar harus mempunyai kemiringan yang sama.<\/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-e5072b7479ca854c5e3cdea8ffff2c0a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=4\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"48\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dalam hal ini mereka tidak memberitahu kita titik singgung antara kurva dan garis singgung. Namun kita mengetahui bahwa turunan kurva pada titik singgung sama dengan kemiringan garis singgung, yaitu<\/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-5c606eb4b268b71562672c32a0461053_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=f'(x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"85\" style=\"vertical-align: -5px;\"><\/p>\n<p> . Nah bagaimana kita bisa mengetahui nilainya<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p> , kita dapat mencari x <sub>0<\/sub> dari persamaan tersebut<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f79089d702fc8e5b6b7342c0eb2f0c68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=f'(x_0):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"95\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Untuk melakukan ini, pertama-tama kita menghitung turunan dari <\/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-40e51e628d64bea41578e16139b71b6a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"43\" 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-ac4abafb3c879a6fd9c906ff9eea94d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)= x^2-2x-1 \\ \\longrightarrow \\ f'(x)=2x-2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"309\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan sekarang kita menyelesaikannya<\/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-5c606eb4b268b71562672c32a0461053_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=f'(x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"85\" style=\"vertical-align: -5px;\"><\/p>\n<p> mengetahui bahwa <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-421352ccc778c624805a5e2663bb7077_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m = 4 :\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"57\" style=\"vertical-align: 0px;\"><\/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-f99f5b23457229b93eb24c214942f41f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m =f'(x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"85\" 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-6a9fc1e268bfa17b6fe04e5fafbaaedc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4 =2(x_0)-2\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"103\" 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-74d7e6af89f911f5a194fee138e70afa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4+2 =2x_0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"89\" style=\"vertical-align: -3px;\"><\/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-96c892854007fa06281252d3fcc3ae4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"6 =2x_0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"59\" style=\"vertical-align: -3px;\"><\/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-9602ea885075b53499286a5126ad9724_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{6}{2} =x_0\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"49\" 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-f2fb028e2d1cb1011436226f865d5162_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3=x_0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"50\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan setelah kita mengetahui koordinat x suatu titik, kita dapat mencari koordinat titik lainnya dengan menghitung <\/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-4763abfc9310baf690c4bb81c5d8b743_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(3):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" 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-9e36063894754424dc75ff41070c42ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(3)=3^2-2\\cdot 3-1= 9-6-1=2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"281\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Jadi, titik yang dilalui kurva dan garis singgung adalah titik<\/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-a3558010337a7cdc27dddb44c10f0df1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(3,2).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"43\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Yang tersisa hanyalah mengganti nilai kemiringan dan titik singgung yang ditemukan ke dalam persamaannya:<\/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-8f1f49e9bef505c5c71cffd15f0d29d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{c} y -y_0= m(x-x_0) \\\\[3ex] m=4 \\qquad P(3,2) \\end{array} \\right\\} \\longrightarrow \\ y -2= 4(x-3)\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"344\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan persamaan garis singgungnya 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-89b6fd61abd22d30db13453334da7135_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y -2=4(x-3)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"126\" 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> Hitung garis singgung kurva 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-ab7fb2898ec2a42b558f032b99518338_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=2x^2+5x+1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"155\" style=\"vertical-align: -5px;\"><\/p>\n<p> yang membentuk sudut 45\u00ba dengan sumbu X. <\/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\"> Rumusan masalah memberitahu kita bahwa garis singgung harus membentuk sudut 45\u00ba terhadap sumbu X. Dalam kasus ini, rumus berikut harus diterapkan untuk mencari nilai kemiringan: <\/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-1e17ba52bc8d7a78aa6abe918856ba28_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m = \\text{tg}\\left(\\alpha\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"82\" 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-c891eec41f7529fbb36d622027b94d46_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m = \\text{tg}\\left(45^{\\text{o}}\\right) = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"129\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Pernyataan tersebut tidak menentukan titik singgung antara kurva dan garis singgung. Namun kita mengetahui bahwa turunan kurva pada titik singgung sama dengan kemiringan garis singgung, yaitu<\/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-5c606eb4b268b71562672c32a0461053_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=f'(x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"85\" style=\"vertical-align: -5px;\"><\/p>\n<p> . Oleh karena itu, kita dapat menghitung x <sub>0<\/sub> dengan menyelesaikan persamaan tersebut<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f79089d702fc8e5b6b7342c0eb2f0c68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=f'(x_0):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"95\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Untuk melakukan ini, pertama-tama kita menghitung turunan dari <\/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-40e51e628d64bea41578e16139b71b6a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"43\" 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-d757979ec817338abf9a0d50e4d8838d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=2x^2+5x+1\\ \\longrightarrow \\ f'(x)=4x+5\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"318\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan sekarang kita menyelesaikannya<\/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-5c606eb4b268b71562672c32a0461053_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=f'(x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"85\" style=\"vertical-align: -5px;\"><\/p>\n<p> mengetahui bahwa <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-32704d9853b0093395b41eb385ebb4e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m = 1 :\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"57\" style=\"vertical-align: 0px;\"><\/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-f99f5b23457229b93eb24c214942f41f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m =f'(x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"85\" 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-eec283acc7af9f75a48ed262d785d7f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"1 =4(x_0)+5\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"102\" 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-4bcad96ea71d673fba2f814bffaee7c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"1-5 =4x_0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"88\" style=\"vertical-align: -3px;\"><\/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-7638e49e3e5eab4f64f4dc439d458ec5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-4 =4x_0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"71\" style=\"vertical-align: -3px;\"><\/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-8561c7a3def54db216a8f1ebf2588e79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{-4}{4} =x_0\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"71\" 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-64d0311d2aa3328e9ed1f2073e90e4bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-1=x_0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"62\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan setelah kita mengetahui koordinat x suatu titik, kita dapat mencari koordinat titik lainnya dengan menghitung <\/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-ab353a75f8c672950ea7d8376104722d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(-1):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"56\" 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-33f491d7f9d0eeeb767d846b5650734f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(-1)=2(-1)^2+5(-1)+1=2\\cdot 1  -5 + 1 = -2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"382\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">Jadi, titik yang dilalui kurva dan garis singgung adalah titik<\/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-d2ae436ead5cc58de912263561cfbe63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(-1,-2).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"70\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Yang tersisa hanyalah mengganti nilai kemiringan dan titik singgung yang ditemukan ke dalam persamaannya:<\/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-8ed772b3993de50c4c67631a6fd33040_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{c} y -y_0= m(x-x_0) \\\\[3ex] m=1 \\qquad P(-1,-2) \\end{array} \\right\\} \\longrightarrow \\ y -(-2)= 1(x-(-1))\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"414\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan terakhir, kita melakukan operasi untuk mencari persamaan garis singgung: <\/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-28409d87564a3385166261f1fe92c01e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y -(-2)=1(x-(-1))\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"182\" 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-9ebfc08d88b0dcf9c22ff7f225afdabf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y +2=1(x+1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"126\" 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-b1c1b9eabdc99bc1ff5b5e8cdb5baf94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y + 2=x+1}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"103\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Pada artikel ini kita akan melihat cara mencari persamaan garis singgung suatu kurva. Selain itu, Anda dapat berlatih dengan latihan yang diselesaikan dengan tingkat kesulitan berbeda. Persamaan garis singgung suatu fungsi di suatu titik Persamaan garis singgung fungsi f(x) di titik x=x 0 adalah: Dimana titik P(x 0 ,y 0 ) merupakan titik dimana garis &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/id\/persamaan-tangen\/\"> <span class=\"screen-reader-text\">Persamaan garis singgung<\/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":[49],"tags":[],"class_list":["post-39","post","type-post","status-publish","format-standard","hentry","category-representasi-fungsi"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Persamaan garis singgung -<\/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\/persamaan-tangen\/\" \/>\n<meta property=\"og:locale\" content=\"id_ID\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Persamaan garis singgung -\" \/>\n<meta property=\"og:description\" content=\"Pada artikel ini kita akan melihat cara mencari persamaan garis singgung suatu kurva. Selain itu, Anda dapat berlatih dengan latihan yang diselesaikan dengan tingkat kesulitan berbeda. 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