[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"public-settings":3,"content:materi:demo-materi-matematika-kuliah-1":11,"public-catalog":153},{"features.tutoring.enabled":4,"sections.subjects.enabled":5,"sections.pricing.enabled":5,"sections.teachers.enabled":5,"sections.testimonials.enabled":4,"sections.faqs.enabled":5,"bankSoal.enabled":5,"site.maintenance":4,"whatsapp.number":6,"brand.name":7,"brand.tagline":8,"contact.email":9,"contact.address":10},false,true,"6285156084435","MyFismat","Platform belajar Matematika, Fisika, dan Pemrograman untuk SD, SMP, SMA, dan Kuliah","support@myfismat.com","Indonesia",{"id":12,"type":13,"slug":14,"title":15,"excerpt":16,"coverUrl":17,"status":18,"publishedAt":19,"featured":5,"subjectId":20,"blocks":21,"seoTitle":17,"seoDescription":17,"readingMinutes":57,"order":58,"createdAt":59,"updatedAt":59,"subject":60,"levels":65,"topics":70,"tags":74,"author":17,"resolved":75,"related":84},"demo-v2-m-matematika-kuliah-1","MATERIAL","demo-materi-matematika-kuliah-1","Turunan sebagai laju sesaat dan pendekatan lokal","Menurunkan fungsi dari limit hasil bagi selisih serta menghubungkan kemiringan, kecepatan, dan garis singgung.",null,"PUBLISHED","2026-10-08T10:03:56.572Z","cmuzbyhdg000471r9l54yc9ia",[22,27,31,35,39,43,47,51],{"id":23,"data":24,"type":26},"section-1",{"markdown":25},"## Tujuan dan prasyarat\n\nSetelah mempelajari materi ini, kamu dapat membedakan laju rata-rata dan laju sesaat, menghitung turunan melalui definisi limit, serta menyusun garis singgung. Prasyaratnya adalah fungsi, limit dasar, pemfaktoran, dan gradien garis. Kita bekerja dengan fungsi real pada interval yang memuat titik pengamatan agar pendekatan dari kiri dan kanan dapat dibandingkan.\n","text",{"id":28,"data":29,"type":26},"section-2",{"markdown":30},"## Dari dua titik menuju satu titik\n\nMisalkan $f(t)$ adalah posisi benda dalam meter dan $t$ waktu dalam sekon. Laju rata-rata antara waktu $a$ dan $a+h$ adalah $\\frac{f(a+h)-f(a)}h$. Pembilang mengukur perubahan posisi, penyebut mengukur perubahan waktu. Satuannya meter per sekon. Pada grafik, rasio ini adalah gradien garis sekan yang melalui dua titik berbeda.\n\nUntuk mendekati laju tepat pada waktu $a$, perkecil interval pengamatan. Jika hasil bagi dari kedua arah menuju nilai sama, limitnya disebut **turunan**: $f'(a)=\\lim_{h\\to0}\\frac{f(a+h)-f(a)}h$. Selama membentuk hasil bagi, $h$ tidak boleh nol. Kita menyederhanakan ekspresi untuk $h\\ne0$, lalu mengambil limit; memasukkan nol langsung ke pecahan awal akan menghasilkan bentuk tidak tentu.\n\nSecara geometri, turunan yang terdefinisi dan berhingga memberikan gradien garis singgung. Secara praktis, ia memberi pendekatan lokal $f(a+h)\\approx f(a)+f'(a)h$ untuk perubahan kecil. Tanda positif menunjukkan kecenderungan naik, tanda negatif turun, dan nol berarti garis singgung mendatar. Turunan nol tidak otomatis membuktikan maksimum atau minimum; misalnya $x^3$ memiliki turunan nol di asal tetapi tetap meningkat melintasi titik itu.\n",{"id":32,"data":33,"type":26},"section-3",{"markdown":34},"## Contoh 1: memperoleh rumus turunan dari definisi\n\nAmbil $f(x)=x^2+3x$.\n\n1. Hitung $f(x+h)=(x+h)^2+3(x+h)=x^2+2xh+h^2+3x+3h$.\n2. Kurangi $f(x)$: tersisa $2xh+h^2+3h$.\n3. Bagi dengan $h\\ne0$: hasil bagi selisih menjadi $2x+h+3$.\n4. Ambil limit $h\\to0$: $f'(x)=2x+3$.\n5. Pada $x=1$, gradiennya lima. Periksa numerik: untuk $h=0{,}01$, gradien sekan 5,01; untuk $h=-0{,}01$, 4,99. Kedua sisi mendekati lima.\n\nKarena $f(1)=4$, garis singgungnya $y-4=5(x-1)$, atau $y=5x-1$. Garis ini melalui titik kurva yang benar dan memiliki gradien hasil limit.\n",{"id":36,"data":37,"type":26},"section-4",{"markdown":38},"## Contoh 2: kecepatan sesaat\n\nPosisi benda diberikan oleh $s(t)=t^2+2t$ meter untuk $t\\ge0$ sekon. Cari kecepatan pada $t=3$ dan bandingkan dengan kecepatan rata-rata pada interval $[3,4]$.\n\n1. Definisi limit memberi $s'(t)=2t+2$ meter per sekon.\n2. Kecepatan sesaat pada tiga sekon adalah $s'(3)=8$ meter per sekon.\n3. Posisi $s(3)=15$ meter dan $s(4)=24$ meter. Kecepatan rata-rata $[24-15]\u002F[4-3]=9$ meter per sekon.\n4. Periksa pada interval lebih pendek $[3,3{,}1]$: posisi akhir 15,81 meter, sehingga laju rata-rata 8,1 meter per sekon, lebih dekat ke delapan.\n5. Pendekatan lokal untuk pertambahan 0,1 sekon memprediksi posisi $15+8(0{,}1)=15{,}8$ meter. Selisih 0,01 meter menunjukkan bahwa pendekatan garis singgung bukan nilai eksak.\n",{"id":40,"data":41,"type":26},"section-5",{"markdown":42},"## Keterbatasan dan kesalahan umum\n\nFungsi yang kontinu belum tentu mempunyai turunan di setiap titik. Pada $f(x)=|x|$, hasil bagi di nol mendekati satu dari kanan dan minus satu dari kiri. Karena berbeda, turunan di nol tidak ada. Jangan pula menyamakan nilai $f(a)$ dengan nilai $f'(a)$: keduanya mengukur hal berbeda dan biasanya mempunyai satuan berbeda.\n",{"id":44,"data":45,"type":26},"section-6",{"markdown":46},"## Ringkasan dan cek mandiri\n\nTurunan adalah limit laju rata-rata, bukan pembagian perubahan dengan interval nol. Untuk $g(x)=x^2$ pada $x=2$, gradien empat dan garis singgung $y=4x-4$. Periksa bahwa garis melalui $(2,4)$, lalu bandingkan prediksi pada $x=2{,}01$ dengan nilai fungsi sebenarnya.",{"id":48,"data":49,"type":26},"sources",{"markdown":50},"## Sumber belajar\n\n- [OpenStax Calculus Volume 1 — Defining the Derivative](https:\u002F\u002Fopenstax.org\u002Fbooks\u002Fcalculus-volume-1\u002Fpages\u002F3-1-defining-the-derivative)\n\nPenjelasan dan contoh disusun ulang untuk materi ini; sumber dapat dibuka untuk belajar lebih lanjut.",{"id":52,"data":53,"type":56},"practice",{"title":54,"questionSetId":55},"Cek pemahaman: Turunan sebagai laju sesaat dan pendekatan 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Kelas 12 Lanjut.","2026-10-08T09:24:41.607Z",{"id":263,"subjectId":20,"levelId":100,"parentId":17,"slug":1755,"name":1756,"description":1757,"order":1758,"createdAt":1759,"updatedAt":1759},"analisis-data-peluang","Analisis Data & Peluang","Analisis Data & Peluang — Kelas 12 Lanjut.",30,"2026-10-08T09:24:41.611Z",{"id":407,"subjectId":159,"levelId":100,"parentId":17,"slug":1761,"name":1762,"description":1763,"order":1758,"createdAt":1764,"updatedAt":1764},"fisika-inti-dan-radioaktivitas","Fisika Inti dan Radioaktivitas","Fisika Inti dan Radioaktivitas — Kelas 12.","2026-10-08T09:24:41.318Z",{"id":1766,"subjectId":20,"levelId":100,"parentId":17,"slug":1767,"name":1768,"description":1769,"order":1770,"createdAt":1771,"updatedAt":1771},"cmuzbyhw700gw71r9o87lq7vd","logaritma","Logaritma","Logaritma — Kelas 12 Lanjut.",31,"2026-10-08T09:24:41.623Z",{"id":1773,"subjectId":20,"levelId":100,"parentId":17,"slug":1774,"name":1775,"description":1776,"order":1777,"createdAt":1778,"updatedAt":1778},"cmuzbyhw900gy71r9d9sjh9ie","program-linier","Program Linier","Program Linier — Kelas 12 Lanjut.",32,"2026-10-08T09:24:41.625Z",{"id":1780,"subjectId":20,"levelId":100,"parentId":17,"slug":1781,"name":1782,"description":1783,"order":1784,"createdAt":1785,"updatedAt":1785},"cmuzbyhwb00h071r97764r7hr","bangun-ruang","Bangun Ruang","Bangun Ruang — Kelas 12 Lanjut.",33,"2026-10-08T09:24:41.627Z",{"id":1787,"subjectId":167,"levelId":17,"parentId":17,"slug":1788,"name":1789,"description":1790,"order":1791,"createdAt":1792,"updatedAt":1792},"demo-v2-topic-pemrograman-sd-1","demo-v2-sd-1","Algoritma: Urutan Perintah yang Jelas","Susun instruksi sederhana, lalu telusuri hasilnya selangkah demi selangkah.",100,"2026-10-08T10:03:56.954Z",{"id":1794,"subjectId":20,"levelId":100,"parentId":17,"slug":1795,"name":1796,"description":1797,"order":1791,"createdAt":1798,"updatedAt":1798},"demo-v2-topic-matematika-sma-1","demo-v2-sma-1","Fungsi kuadrat: akar, puncak, dan bentuk grafik","Menghubungkan bentuk umum, faktor, dan kuadrat sempurna untuk memahami parabola dan nilai optimum.","2026-10-08T10:03:56.485Z",{"id":1800,"subjectId":167,"levelId":100,"parentId":17,"slug":1795,"name":1801,"description":1802,"order":1791,"createdAt":1803,"updatedAt":1803},"demo-v2-topic-pemrograman-sma-1","Fungsi: Memisahkan Masukan, Proses, dan Hasil","Buat fungsi Python yang bisa dipakai ulang dan bedakan return dari print.","2026-10-08T10:03:57.108Z",{"id":1805,"subjectId":159,"levelId":96,"parentId":17,"slug":1806,"name":1807,"description":1808,"order":1791,"createdAt":1809,"updatedAt":1809},"demo-v2-topic-fisika-smp-1","demo-v2-smp-1","Gerak Lurus: Jarak, Perpindahan, dan Grafik Posisi","Membedakan kelajuan dan kecepatan rata-rata serta membaca arti kemiringan grafik posisi terhadap waktu.","2026-10-08T10:03:56.729Z",{"id":1811,"subjectId":159,"levelId":100,"parentId":17,"slug":1795,"name":1812,"description":1813,"order":1791,"createdAt":1814,"updatedAt":1814},"demo-v2-topic-fisika-sma-1","Hukum II Newton dan Gesekan pada Bidang Datar","Membuat diagram gaya, menentukan resultan, dan membedakan gesekan statis dari gesekan kinetik.","2026-10-08T10:03:56.801Z",{"id":1816,"subjectId":159,"levelId":17,"parentId":17,"slug":1788,"name":1817,"description":1818,"order":1791,"createdAt":1819,"updatedAt":1819},"demo-v2-topic-fisika-sd-1","Mengukur Panjang dan Volume Benda","Membaca penggaris, memakai satuan baku, dan menghitung isi kubus melalui susunan kubus kecil.","2026-10-08T10:03:56.657Z",{"id":1821,"subjectId":167,"levelId":67,"parentId":17,"slug":73,"name":1822,"description":1823,"order":1791,"createdAt":1824,"updatedAt":1824},"demo-v2-topic-pemrograman-kuliah-1","Pencarian Linear, Set, dan Biaya Operasi","Bandingkan cara mencari anggota data melalui langkah yang benar-benar dikerjakan.","2026-10-08T10:03:57.182Z",{"id":1826,"subjectId":20,"levelId":96,"parentId":17,"slug":1806,"name":1827,"description":1828,"order":1791,"createdAt":1829,"updatedAt":1829},"demo-v2-topic-matematika-smp-1","Perbandingan senilai dan berbalik nilai","Memilih model dari besaran yang tetap: rasio tetap untuk senilai dan hasil kali tetap untuk berbalik nilai.","2026-10-08T10:03:56.412Z",{"id":72,"subjectId":20,"levelId":67,"parentId":17,"slug":73,"name":15,"description":16,"order":1791,"createdAt":1831,"updatedAt":1831},"2026-10-08T10:03:56.556Z",{"id":1833,"subjectId":167,"levelId":96,"parentId":17,"slug":1806,"name":1834,"description":1835,"order":1791,"createdAt":1836,"updatedAt":1836},"demo-v2-topic-pemrograman-smp-1","Variabel, Tipe Data, dan Keluaran Python","Simpan nilai dengan nama yang jelas dan bedakan angka dari teks.","2026-10-08T10:03:57.035Z",{"id":1838,"subjectId":159,"levelId":67,"parentId":17,"slug":73,"name":1839,"description":1840,"order":1791,"createdAt":1841,"updatedAt":1841},"demo-v2-topic-fisika-kuliah-1","Vektor Tiga Dimensi dan Usaha sebagai Hasil Kali Titik","Menghubungkan komponen, panjang vektor, vektor satuan, proyeksi, dan makna fisik hasil kali titik.","2026-10-08T10:03:56.874Z",{"id":1843,"subjectId":20,"levelId":100,"parentId":17,"slug":1844,"name":1845,"description":1846,"order":1847,"createdAt":1848,"updatedAt":1848},"demo-v2-topic-matematika-sma-2","demo-v2-sma-2","Eksponen dan logaritma: mencari pangkat yang belum diketahui","Membedakan kenaikan tetap dan faktor tetap, lalu membaca logaritma sebagai operasi balik eksponen.",101,"2026-10-08T10:03:56.507Z",{"id":1850,"subjectId":20,"levelId":96,"parentId":17,"slug":1851,"name":1852,"description":1853,"order":1847,"createdAt":1854,"updatedAt":1854},"demo-v2-topic-matematika-smp-2","demo-v2-smp-2","Fungsi linear: gradien dan titik potong","Membaca y = mx + b sebagai perubahan tetap, menghitung gradien dari dua titik, dan memeriksa grafik.","2026-10-08T10:03:56.434Z",{"id":1856,"subjectId":159,"levelId":17,"parentId":17,"slug":1857,"name":1858,"description":1859,"order":1847,"createdAt":1860,"updatedAt":1860},"demo-v2-topic-fisika-sd-2","demo-v2-sd-2","Gaya: Dorongan, Tarikan, dan Perubahan Gerak","Mengenali siapa yang memberikan gaya, arah gaya, dan perubahan gerak atau bentuk yang dapat diamati.","2026-10-08T10:03:56.679Z",{"id":1862,"subjectId":159,"levelId":100,"parentId":17,"slug":1844,"name":1863,"description":1864,"order":1847,"createdAt":1865,"updatedAt":1865},"demo-v2-topic-fisika-sma-2","Gerak Peluru: Dua Arah dengan Satu Waktu","Menguraikan kecepatan awal, menghitung waktu terbang, tinggi maksimum, dan jangkauan untuk model ideal.","2026-10-08T10:03:56.824Z",{"id":135,"subjectId":20,"levelId":67,"parentId":17,"slug":136,"name":125,"description":126,"order":1847,"createdAt":1867,"updatedAt":1867},"2026-10-08T10:03:56.578Z",{"id":1869,"subjectId":167,"levelId":67,"parentId":17,"slug":136,"name":1870,"description":1871,"order":1847,"createdAt":1872,"updatedAt":1872},"demo-v2-topic-pemrograman-kuliah-2","Kelas dan Invarian pada Objek","Kelola keadaan dan perilaku bersama, lalu jaga agar objek selalu valid.","2026-10-08T10:03:57.206Z",{"id":1874,"subjectId":167,"levelId":100,"parentId":17,"slug":1844,"name":1875,"description":1876,"order":1847,"createdAt":1877,"updatedAt":1877},"demo-v2-topic-pemrograman-sma-2","List dan Dictionary untuk Mengolah Data","Pilih struktur data sesuai cara mengaksesnya dan hitung ringkasan data.","2026-10-08T10:03:57.131Z",{"id":1879,"subjectId":159,"levelId":96,"parentId":17,"slug":1851,"name":1880,"description":1881,"order":1847,"createdAt":1882,"updatedAt":1882},"demo-v2-topic-fisika-smp-2","Massa Jenis: Menghubungkan Massa dan Volume","Menghitung massa jenis, mengukur volume dengan perpindahan air, dan menjaga konsistensi satuan.","2026-10-08T10:03:56.750Z",{"id":1884,"subjectId":159,"levelId":67,"parentId":17,"slug":136,"name":1885,"description":1886,"order":1847,"createdAt":1887,"updatedAt":1887},"demo-v2-topic-fisika-kuliah-2","Osilasi Harmonik Sederhana: Fase, Turunan, dan Energi","Menganalisis model x = A cos(ωt), menurunkan kecepatan dan percepatan, serta memeriksa kekekalan energi.","2026-10-08T10:03:56.896Z",{"id":1889,"subjectId":167,"levelId":17,"parentId":17,"slug":1857,"name":1890,"description":1891,"order":1847,"createdAt":1892,"updatedAt":1892},"demo-v2-topic-pemrograman-sd-2","Pengulangan: Langkah yang Dikerjakan Berkali-kali","Kenali pola berulang dan hitung hasil tiap putaran tanpa kehilangan jejak.","2026-10-08T10:03:56.978Z",{"id":1894,"subjectId":167,"levelId":96,"parentId":17,"slug":1851,"name":1895,"description":1896,"order":1847,"createdAt":1897,"updatedAt":1897},"demo-v2-topic-pemrograman-smp-2","Percabangan Python dan Batas Keputusan","Tulis if, elif, dan else serta uji nilai di sekitar batas.","2026-10-08T10:03:57.057Z",{"id":1899,"subjectId":159,"levelId":100,"parentId":17,"slug":151,"name":1900,"description":1901,"order":1902,"createdAt":1903,"updatedAt":1903},"demo-v2-topic-fisika-sma-3","Hukum Ohm, Daya, dan Energi pada Resistor","Menghubungkan tegangan, arus, hambatan, dan energi listrik tanpa mencampurkan satuan volt, ampere, watt, dan joule.",102,"2026-10-08T10:03:56.846Z",{"id":1905,"subjectId":167,"levelId":100,"parentId":17,"slug":151,"name":1906,"description":1907,"order":1902,"createdAt":1908,"updatedAt":1908},"demo-v2-topic-pemrograman-sma-3","Membaca Berkas dan Menangani Masukan Salah","Ubah baris teks menjadi angka dan tangani kesalahan yang bisa diperkirakan.","2026-10-08T10:03:57.153Z",{"id":150,"subjectId":20,"levelId":100,"parentId":17,"slug":151,"name":141,"description":142,"order":1902,"createdAt":1910,"updatedAt":1910},"2026-10-08T10:03:56.528Z",{"id":1912,"subjectId":167,"levelId":67,"parentId":17,"slug":120,"name":1913,"description":1914,"order":1902,"createdAt":1915,"updatedAt":1915},"demo-v2-topic-pemrograman-kuliah-3","Pengujian Otomatis dan Kasus Batas","Uji kontrak fungsi dengan contoh biasa, batas, dan masukan tidak sah.","2026-10-08T10:03:57.229Z",{"id":1917,"subjectId":167,"levelId":17,"parentId":17,"slug":1918,"name":1919,"description":1920,"order":1902,"createdAt":1921,"updatedAt":1921},"demo-v2-topic-pemrograman-sd-3","demo-v2-sd-3","Percabangan: Memilih Tindakan dengan Syarat","Gunakan pertanyaan ya atau tidak untuk menentukan tindakan yang tepat.","2026-10-08T10:03:57.006Z",{"id":1923,"subjectId":167,"levelId":96,"parentId":17,"slug":1924,"name":1925,"description":1926,"order":1902,"createdAt":1927,"updatedAt":1927},"demo-v2-topic-pemrograman-smp-3","demo-v2-smp-3","Perulangan for dan Akumulator","Hitung total data dengan jejak putaran yang dapat diperiksa.","2026-10-08T10:03:57.079Z",{"id":1929,"subjectId":159,"levelId":17,"parentId":17,"slug":1918,"name":1930,"description":1931,"order":1902,"createdAt":1932,"updatedAt":1932},"demo-v2-topic-fisika-sd-3","Suhu dan Perpindahan Kalor di Sekitar Kita","Membaca perubahan suhu dan membedakan konduksi, konveksi, serta radiasi melalui kejadian sehari-hari.","2026-10-08T10:03:56.700Z",{"id":1934,"subjectId":20,"levelId":96,"parentId":17,"slug":1924,"name":1935,"description":1936,"order":1902,"createdAt":1937,"updatedAt":1937},"demo-v2-topic-matematika-smp-3","Teorema Pythagoras dan jarak diagonal","Menentukan sisi segitiga siku-siku dengan kuadrat panjang, lalu memeriksa hasil melalui geometri dan satuan.","2026-10-08T10:03:56.456Z",{"id":1939,"subjectId":159,"levelId":67,"parentId":17,"slug":120,"name":1940,"description":1941,"order":1902,"createdAt":1942,"updatedAt":1942},"demo-v2-topic-fisika-kuliah-3","Usaha Gaya Berubah: Integral dan Teorema Energi Kinetik","Menghitung luas bertanda di bawah grafik gaya, menghubungkannya dengan integral, dan menentukan kelajuan akhir.","2026-10-08T10:03:56.918Z",{"id":1944,"subjectId":159,"levelId":96,"parentId":17,"slug":1924,"name":1945,"description":1946,"order":1902,"createdAt":1947,"updatedAt":1947},"demo-v2-topic-fisika-smp-3","Usaha dan Energi: Memeriksa Perubahan dengan Joule","Menghitung usaha gaya searah atau berlawanan, energi gerak, dan perubahan energi ketika benda diangkat.","2026-10-08T10:03:56.772Z",{"id":119,"subjectId":20,"levelId":67,"parentId":17,"slug":120,"name":109,"description":110,"order":1902,"createdAt":1949,"updatedAt":1949},"2026-10-08T10:03:56.615Z",[]]