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Eğitmen, önceki videoda kosinüs x'in Maclaurin serisini bulduklarını hatırlatarak, bu videoda sinüs x'in Maclaurin serisini hesaplamayı göstermektedir.","Videoda, sinüs x'in türevlerinin hesaplanması, Maclaurin serisinin formülünün uygulanması ve sinüs x'in polinom gösterimi detaylı olarak anlatılmaktadır. Eğitmen, sinüs x'in Maclaurin serisinin kosinüs x'in Maclaurin serisiyle karşılaştırmasını yaparak, sinüs x'in polinom gösteriminde x'in tek üslü ifadelerinin ve işaretlerin nasıl değiştiğini açıklamaktadır. Video, bir sonraki derste e üzeri x'in Maclaurin serisinin hesaplanacağını belirterek sona ermektedir."]},"endTime":298,"title":"Sinüs X'in Maclaurin Serisi","beginTime":0}],"fullResult":[{"index":0,"title":"Sinüs x'in Maclaurin Serisi","list":{"type":"unordered","items":["Önceki videoda kosinüs x'in Maclaurin serisi bulunmuş ve yaklaşık değerler elde edilmişti.","Bu videoda sinüs x'in Maclaurin serisi bulunacak ve benzer bir örüntü olup olmadığı incelenecek.","Maclaurin serisi, ortalanmış bir Taylor serisidir."]},"beginTime":0,"endTime":29,"href":"/video/preview/14493929936499741042?parent-reqid=1769778737582064-10932758592689942081-balancer-l7leveler-kubr-yp-sas-212-BAL&text=0X_InRey&t=0&ask_summarization=1"},{"index":1,"title":"Sinüs x'in Türevleri","list":{"type":"unordered","items":["Sinüs x'in türevleri: f'(x) = sinüs x, f''(x) = eksi sinüs x, f'''(x) = eksi kosinüs x, f''''(x) = sinüs x şeklinde bir döngüye girmektedir.","f(0) = 0, f'(0) = 1, f''(0) = 0, f'''(0) = eksi 1, f''''(0) = 1 şeklinde bir örüntü oluşmaktadır.","Bu örüntü, sinüs ve kosinüs fonksiyonlarının türevlerinin 0'daki değerlerini göstermektedir."]},"beginTime":29,"endTime":110,"href":"/video/preview/14493929936499741042?parent-reqid=1769778737582064-10932758592689942081-balancer-l7leveler-kubr-yp-sas-212-BAL&text=0X_InRey&t=29&ask_summarization=1"},{"index":2,"title":"Sinüs x'in Maclaurin Serisi","list":{"type":"unordered","items":["Sinüs x'in Maclaurin serisi, kosinüs x'in Maclaurin serisine benzer şekilde yazılabilir.","Sinüs x'in Maclaurin serisi: x + x³/3! + x⁵/5! + ... şeklinde devam eder.","Sinüs x'in Maclaurin serisinde x'in tek üsleri bulunur ve işaretler artı-eksi şeklinde değişir."]},"beginTime":110,"endTime":259,"href":"/video/preview/14493929936499741042?parent-reqid=1769778737582064-10932758592689942081-balancer-l7leveler-kubr-yp-sas-212-BAL&text=0X_InRey&t=110&ask_summarization=1"},{"index":3,"title":"Kosinüs ve Sinüs Arasındaki 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fazlası için: http://www.khanacademy.org.tr Matematikten sanat tarihine, ekonomiden fen bilimlerine, basit toplamadan diferansiyel denklemlere, ilkokul seviyesinden üniversite seviyesine her...","preview":{"posterSrc":"//avatars.mds.yandex.net/get-vthumb/4303409/ef0af8014940ea8ff8020f8676f4af40/564x318_1","videoSrc":"https://video-preview.s3.yandex.net/nHHFGAAAAAA.mp4","videoType":"video/mp4"},"target":"_self","position":"9","reqid":"1769778737582064-10932758592689942081-balancer-l7leveler-kubr-yp-sas-212-BAL","summary":{"isFull":true,"fullTextUrl":"/video/result?ask_summarization=1&numdoc=1&noreask=1&nomisspell=1&parent-reqid=1769778737582064-10932758592689942081-balancer-l7leveler-kubr-yp-sas-212-BAL&text=videoid:16577988393500172895","teaser":[{"list":{"type":"unordered","items":["Bu video, bir matematik eğitim içeriğidir. Eğitmen, f(x) = ln√x fonksiyonunun türevini bulma yöntemini adım adım anlatmaktadır.","Videoda, f(x) fonksiyonunun iki fonksiyonun bileşkesi olduğu açıklanarak zincir kuralı kullanılarak türevinin nasıl alınacağı gösterilmektedir. Eğitmen önce fonksiyonun bileşke yapısını şemalarla açıklar, ardından kuvvet kuralı ve doğal logaritmanın türevi bilgilerini kullanarak f'(x) = 1/2√x sonucunu elde eder. Video, bu tür sorularda pratik yapmanın önemini vurgulayarak, daha sonra şemaları çizmeden de bu tür soruları çözebileceğimizi belirtir."]},"endTime":321,"title":"Karekök İçinde X'in Doğal Logaritmasının Türevi","beginTime":0}],"fullResult":[{"index":0,"title":"Fonksiyonun Bileşke Yapısı","list":{"type":"unordered","items":["f(x) = ln(√x) fonksiyonu, iki farklı fonksiyonun bileşkesi olarak yazılabilir.","Fonksiyon, önce x'in karekökünü alıyor (u(x) = √x), sonra bu karekökün doğal logaritmasını alıyor (v(x) = ln(x)).","f(x) = v(u(x)) şeklinde ifade edilebilir, yani f(x) = ln(√x)."]},"beginTime":0,"endTime":132,"href":"/video/preview/16577988393500172895?parent-reqid=1769778737582064-10932758592689942081-balancer-l7leveler-kubr-yp-sas-212-BAL&text=0X_InRey&t=0&ask_summarization=1"},{"index":1,"title":"Zincir Kuralının Uygulanması","list":{"type":"unordered","items":["Fonksiyonun bileşke yapısı anlaşıldığında, türevini almak için zincir kuralı kullanılabilir.","Zincir kuralı: f'(x) = v'(u(x)) × u'(x) şeklindedir.","u'(x) = 1/2√x ve v'(x) = 1/x olarak hesaplanır."]},"beginTime":132,"endTime":199,"href":"/video/preview/16577988393500172895?parent-reqid=1769778737582064-10932758592689942081-balancer-l7leveler-kubr-yp-sas-212-BAL&text=0X_InRey&t=132&ask_summarization=1"},{"index":2,"title":"Türevin Hesaplanması","list":{"type":"unordered","items":["v'(u(x)) = 1/u(x) = 1/√x olarak bulunur.","f'(x) = (1/2√x) × (1/√x) = 1/2 olarak hesaplanır.","Bu tür sorularda pratik yaparak, şemaları çizmeden de çözüm yapılabilir."]},"beginTime":199,"endTime":317,"href":"/video/preview/16577988393500172895?parent-reqid=1769778737582064-10932758592689942081-balancer-l7leveler-kubr-yp-sas-212-BAL&text=0X_InRey&t=199&ask_summarization=1"}],"linkTemplate":"/video/preview/16577988393500172895?parent-reqid=1769778737582064-10932758592689942081-balancer-l7leveler-kubr-yp-sas-212-BAL&text=0X_InRey&t=%%timestamp%%&ask_summarization=1"},"isAdultDoc":false,"relatedParams":{"text":"Örnek: ln( x)’in Türevini Zincir Kuralını Kullanarak Alma 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fazlası için: http://www.khanacademy.org.tr Matematikten sanat tarihine, ekonomiden fen bilimlerine, basit toplamadan diferansiyel denklemlere, ilkokul seviyesinden üniversite seviyesine...","preview":{"posterSrc":"//avatars.mds.yandex.net/get-vthumb/2218504/b4a5eea0f7d5e3b4ad40aae9d153c55b/564x318_1","videoSrc":"https://video-preview.s3.yandex.net/wnr7WQAAAAA.mp4","videoType":"video/mp4"},"target":"_self","position":"10","reqid":"1769778737582064-10932758592689942081-balancer-l7leveler-kubr-yp-sas-212-BAL","summary":{"isFull":true,"fullTextUrl":"/video/result?ask_summarization=1&numdoc=1&noreask=1&nomisspell=1&parent-reqid=1769778737582064-10932758592689942081-balancer-l7leveler-kubr-yp-sas-212-BAL&text=videoid:3286237784549626137","teaser":[{"list":{"type":"unordered","items":["Bu video, matematik eğitimi formatında sunulan bir ders anlatımıdır. Konuşmacı, sıklıkla karşılaşılan bazı fonksiyonların türevlerini listeleyerek izleyicilere aktarmaktadır.","Video, trigonometrik fonksiyonların (sin x, cos x, tan x) türevlerini, ardından üstel ve logaritmik fonksiyonların (e^x, ln x) türevlerini açıklamaktadır. Konuşmacı, bu türevleri kanıtlamak yerine sadece bilinmesi gereken sonuçları vererek, özellikle e^x'in türevinin kendisine eşit olması gibi ilginç özellikleri vurgulamaktadır. Video, ilerideki matematik problemlerinde bu bilgilerin işe yarayacağını belirterek sonlanmaktadır."]},"endTime":241,"title":"Sıklıkla Karşılaşılan Fonksiyonların Türevleri","beginTime":0}],"fullResult":[{"index":0,"title":"Trigonometrik Fonksiyonların Türevleri","list":{"type":"unordered","items":["Sinüs x'in türevi kosinüs x'tir.","Kosinüs x'in türevi eksi sinüs x'tir.","Tanjant x'in türevi bir bölü kosinüs kare x (sekant kare x)'e eşittir."]},"beginTime":10,"endTime":75,"href":"/video/preview/3286237784549626137?parent-reqid=1769778737582064-10932758592689942081-balancer-l7leveler-kubr-yp-sas-212-BAL&text=0X_InRey&t=10&ask_summarization=1"},{"index":1,"title":"Üstel Fonksiyonların Türevleri","list":{"type":"unordered","items":["e üzeri x'in x'e göre türevi e üzeri x'tir.","e üzeri x fonksiyonunun her noktada teğetin eğimi, fonksiyonun o noktadaki değerine eşittir.","Bu özellik e sayısının matematikteki özel konumunu oluşturur."]},"beginTime":75,"endTime":189,"href":"/video/preview/3286237784549626137?parent-reqid=1769778737582064-10932758592689942081-balancer-l7leveler-kubr-yp-sas-212-BAL&text=0X_InRey&t=75&ask_summarization=1"},{"index":2,"title":"Logaritmik Fonksiyonların Türevleri","list":{"type":"unordered","items":["ln x'in türevi bir bölü x (x üzeri eksi bir)dir.","Tüm fonksiyonlar arasında türevi x üzeri eksi bir olan sadece ln x'tir.","Bu videoda türevlerin nasıl bulunduğundan değil, sık karşılaşılan fonksiyonların türevlerinin listelenmesinden bahsedilmektedir."]},"beginTime":189,"endTime":237,"href":"/video/preview/3286237784549626137?parent-reqid=1769778737582064-10932758592689942081-balancer-l7leveler-kubr-yp-sas-212-BAL&text=0X_InRey&t=189&ask_summarization=1"}],"linkTemplate":"/video/preview/3286237784549626137?parent-reqid=1769778737582064-10932758592689942081-balancer-l7leveler-kubr-yp-sas-212-BAL&text=0X_InRey&t=%%timestamp%%&ask_summarization=1"},"isAdultDoc":false,"relatedParams":{"text":"sin x, cos x, tan x, e^x ve ln x'in Türevleri (Kalkülüs / Türev 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üniversite matematiği derslerinden olasılık ve istatistik dersine ait \" F(x)'i Oluşturan f(x)'in Kesikli mi Sürekli mi Olduğunu Belirleme\" videosudur. Hazırlayan: Kemal Duran (Matematik...","preview":{"posterSrc":"//avatars.mds.yandex.net/get-vthumb/979663/bada03e826a2946215c69c560bde6d03/564x318_1","videoSrc":"https://video-preview.s3.yandex.net/rFrd5gAAAAA.mp4","videoType":"video/mp4"},"target":"_self","position":"12","reqid":"1769778737582064-10932758592689942081-balancer-l7leveler-kubr-yp-sas-212-BAL","summary":{"isFull":true,"fullTextUrl":"/video/result?ask_summarization=1&numdoc=1&noreask=1&nomisspell=1&parent-reqid=1769778737582064-10932758592689942081-balancer-l7leveler-kubr-yp-sas-212-BAL&text=videoid:1157867608769959825","teaser":[{"list":{"type":"unordered","items":["Bu video, olasılık ve istatistik derslerinde birikimli dağılım fonksiyonunu (büyük FX) oluşturan küçük FX'in (kesikli veya sürekli olasılık dağılımı) nasıl belirleneceğini anlatan bir eğitim içeriğidir.","Video, büyük FX'in kesikli veya sürekli olasılık dağılımından oluştuğunu belirleme yöntemlerini açıklamaktadır. Kesikli FX'in tüm değerlerinin sayı olduğu, sürekli FX'in ise içinde x'li ifadeler bulunduğu örneklerle gösterilmektedir. Video, geriye dönüş konusuna geçiş yaparak, bir sonraki videoda büyük FX'ten küçük FX'e nasıl dönüş yapılacağını anlatacağını belirtmektedir."]},"endTime":402,"title":"Büyük FX'in Kesikli mi Sürekli mi Olduğunu Belirleme","beginTime":0}],"fullResult":[{"index":0,"title":"Büyük Fx'in Oluşumunu Belirleme","list":{"type":"unordered","items":["Büyük Fx, olasılık-istatistik derslerinde birikimli dağılım fonksiyonunu temsil eder.","Birikimli dağılım fonksiyonu (küçük Fx) ya kesikli olasılık dağılımı (olasılık kütle fonksiyonu) ya da sürekli olasılık dağılımı (olasılık yoğunluk fonksiyonu) olabilir.","Fx verildiğinde, bu Fx'in kesikli olasılık dağılımından mı yoksa sürekli olasılık dağılımından mı oluşturulmuş olduğunu belirlemek, geriye dönüş işlemi için 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Bir eğitmen, sekant ve kosekant fonksiyonlarının türevlerini bulma konusunu açıklamaktadır.","Videoda, önceki videoda tanjant ve kotanjant fonksiyonlarının türevlerinin bölüm kuralı kullanılarak nasıl hesaplandığı hatırlatılarak başlanıyor. Ardından sekant x'in türevi (sinüs x bölü kosinüs kare veya tanjant x çarpı sekant x) ve kosekant x'in türevi (eksi kosinüs x bölü sinüs kare veya eksi kotanjant x çarpı kosekant x) bölüm kuralı kullanılarak detaylı olarak hesaplanıyor. Eğitmen, sekant ve kosekant fonksiyonlarının türevlerini ezberlemek için \"sekant s ile kosinüs k ile başlar, kosekant k ile sinüs de s ile başlar\" gibi bir hatırlatma yöntemi sunuyor."]},"endTime":222,"title":"Sekant ve Kosekant Fonksiyonlarının Türevleri","beginTime":0}],"fullResult":[{"index":0,"title":"Sekant x'in Türevi","list":{"type":"unordered","items":["Önceki videoda tanjant x ve kotanjant x'in türevleri bölüm kuralı kullanılarak bulundu.","Bu videoda sekant x ve kosekant x'in türevleri aynı mantıkla hesaplanacak.","Sekant x, 1/cos x şeklinde yazılabilir ve türevi almak için bölüm kuralı kullanılabilir."]},"beginTime":0,"endTime":48,"href":"/video/preview/2042901616076391257?parent-reqid=1769778737582064-10932758592689942081-balancer-l7leveler-kubr-yp-sas-212-BAL&text=0X_InRey&t=0&ask_summarization=1"},{"index":1,"title":"Sekant x'in Türevinin Hesaplanması","list":{"type":"unordered","items":["Bölüm kuralı uygulanarak sekant x'in türevi: (0 × cos x - 1 × (-sin x)) / cos²x = sin x / cos²x olarak bulunur.","Sekant x'in türevi sin x / cos²x veya tan x × sec x şeklinde ifade edilebilir."]},"beginTime":48,"endTime":123,"href":"/video/preview/2042901616076391257?parent-reqid=1769778737582064-10932758592689942081-balancer-l7leveler-kubr-yp-sas-212-BAL&text=0X_InRey&t=48&ask_summarization=1"},{"index":2,"title":"Kosekant x'in Türevi","list":{"type":"unordered","items":["Kosekant x, 1/sin x şeklinde yazılabilir ve türevi almak için bölüm kuralı kullanılabilir.","Kosekant x'in türevi: (0 × sin x - 1 × cos x) / sin²x = -cos x / sin²x olarak bulunur.","Kosekant x'in türevi -cos x / sin x × 1/sin x veya -cot x × csc x şeklinde ifade edilebilir."]},"beginTime":123,"endTime":215,"href":"/video/preview/2042901616076391257?parent-reqid=1769778737582064-10932758592689942081-balancer-l7leveler-kubr-yp-sas-212-BAL&text=0X_InRey&t=123&ask_summarization=1"}],"linkTemplate":"/video/preview/2042901616076391257?parent-reqid=1769778737582064-10932758592689942081-balancer-l7leveler-kubr-yp-sas-212-BAL&text=0X_InRey&t=%%timestamp%%&ask_summarization=1"},"isAdultDoc":false,"relatedParams":{"text":"sec(x) ve csc(x)'in Türevleri (Matematik) 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Karmaşık (Complex) Analizde çok fazla karşımıza çıkan bir ifadeydi.","preview":{"posterSrc":"//avatars.mds.yandex.net/get-vthumb/3445373/9878139b572f678c546a8cf000e47ff6/564x318_1","videoSrc":"https://video-preview.s3.yandex.net/f6lA0AEAAAA.mp4","videoType":"video/mp4"},"target":"_self","position":"19","reqid":"1769778737582064-10932758592689942081-balancer-l7leveler-kubr-yp-sas-212-BAL","summary":{"isFull":true,"fullTextUrl":"/video/result?ask_summarization=1&numdoc=1&noreask=1&nomisspell=1&parent-reqid=1769778737582064-10932758592689942081-balancer-l7leveler-kubr-yp-sas-212-BAL&text=videoid:2771589458278181586","teaser":[{"list":{"type":"unordered","items":["Bu video, bir matematik eğitmeni tarafından sunulan matematiksel bir ders formatındadır. Eğitmen, Instagram'dan gelen bir istek üzerine e^x fonksiyonunun türevi ve Euler eşitliği konusunu anlatmaktadır.","Video, e^x fonksiyonunun türevinin her derecesinin kendisine eşit olduğunu göstererek başlamakta, ardından Taylor seri açılımlarının temel prensiplerini açıklamaktadır. Eğitmen, e^x fonksiyonunun Taylor serisi açılımını, cosinüs ve sinüs fonksiyonlarının Taylor serisi açılımlarını kullanarak e^(ix) ifadesinin cos(x) + i*sin(x) olduğunu kanıtlamaktadır. Son olarak, e^(iπ) = -1 formülünün fiziksel yorumu yapılarak, karmaşık sayıların vektör olarak gösterilmesi ve bu formülün matematiksel önemi açıklanmaktadır.","Videoda Desmos sitesi kullanılarak teoriler görsel olarak doğrulanmakta ve karmaşık sayılar, türev ve kompleks analiz konularına değinilmektedir. İçerik, 12. sınıf seviyesinden sonra öğrenilebilecek bir matematik dersi niteliğindedir."]},"endTime":1355,"title":"E^x Fonksiyonunun Türevi ve Euler Eşitliği Dersi","beginTime":0}],"fullResult":[{"index":0,"title":"E^π+1 Eşitliği ve Matematiksel Önemi","list":{"type":"unordered","items":["Video, Instagram'da gelen isteklere cevap olarak, matematikte önemli olan e^π+1 eşitliğini açıklamayı amaçlıyor.","Bu eşitlik, matematiğin önemli sayılarını (e, π, i) bir arada topladığı için ilginçtir.","Tabanı pozitif olan bir sayının her kuvveti yine pozitif olmasına rağmen, bu eşitlik ilginç bir sonuç veriyor."]},"beginTime":0,"endTime":61,"href":"/video/preview/2771589458278181586?parent-reqid=1769778737582064-10932758592689942081-balancer-l7leveler-kubr-yp-sas-212-BAL&text=0X_InRey&t=0&ask_summarization=1"},{"index":1,"title":"Eşitliğin Matematiksel Temeli","list":{"type":"unordered","items":["Konuşmacı, bu eşitlikle üçüncü sınıfta tanıştığını ve kompleks analiz dersinde öğrendiğini belirtiyor.","Kompleks düzlemde ve kompleks sayılarla yapılan işlemlerde, reel sayılarla yapılanlara göre farklı sonuçlar elde edilebilir.","Örneğin, sinüs fonksiyonuna imajiner sayılar koyduğumuzda, beklenmedik sonuçlar elde edilebilir."]},"beginTime":61,"endTime":119,"href":"/video/preview/2771589458278181586?parent-reqid=1769778737582064-10932758592689942081-balancer-l7leveler-kubr-yp-sas-212-BAL&text=0X_InRey&t=61&ask_summarization=1"},{"index":2,"title":"E^x Fonksiyonunun Yaklaşımı","list":{"type":"unordered","items":["E^x fonksiyonuna yaklaşıp yaklaşamama problemi, polinomlar kullanılarak çözülebilir.","Sıfırıncı dereceden polinom (sabit fonksiyon) için, e^x fonksiyonunu bir noktasından geçiren fonksiyon y=1 olur.","Birinci dereceden polinom için, e^x fonksiyonunu bir noktasından geçiren ve teğet olan fonksiyon y=x+1 olur."]},"beginTime":119,"endTime":256,"href":"/video/preview/2771589458278181586?parent-reqid=1769778737582064-10932758592689942081-balancer-l7leveler-kubr-yp-sas-212-BAL&text=0X_InRey&t=119&ask_summarization=1"},{"index":3,"title":"Yüksek Dereceli Polinomlarla Yaklaşım","list":{"type":"unordered","items":["İkinci dereceden polinom için, e^x fonksiyonunu bir noktasından geçiren ve teğet olan fonksiyon y=1/2x²+x+1 olur.","Her dereceden türevi alındığında, e^x fonksiyonu kendisine eşit olduğu için, polinomların katsayıları belirlenebilir.","Daha yüksek dereceli polinomlar kullanıldıkça, e^x fonksiyonuna daha yakın yaklaşılır."]},"beginTime":256,"endTime":463,"href":"/video/preview/2771589458278181586?parent-reqid=1769778737582064-10932758592689942081-balancer-l7leveler-kubr-yp-sas-212-BAL&text=0X_InRey&t=256&ask_summarization=1"},{"index":4,"title":"Taylor Serisi Açılımı","list":{"type":"unordered","items":["Taylor serisi açılımı, bir fonksiyonu polinomlarla gösterme yöntemidir ve sadece sıfırdaki gösterimle ilgilenir.","Her fonksiyonun Taylor serisi açılımını alırken keyfi bir noktada alamazsınız, yakınsaklıklar önemlidır.","e üzeri x fonksiyonu, polinomlarla gösterildiğinde kuvvetlerin paydasında faktöriyel değerleri bulunur (örneğin 5'in paydası 120, 4'ün paydası 24)."]},"beginTime":465,"endTime":606,"href":"/video/preview/2771589458278181586?parent-reqid=1769778737582064-10932758592689942081-balancer-l7leveler-kubr-yp-sas-212-BAL&text=0X_InRey&t=465&ask_summarization=1"},{"index":5,"title":"Taylor Serisi Formülü","list":{"type":"unordered","items":["Taylor serisi formülü: f(x) = Σ (f^(n)(a) / n!) * (x-a)^n'dir, burada n=0'dan sonsuza kadar devam eder.","e üzeri x'in tüm türevleri 1'e eşittir, bu nedenle e üzeri x fonksiyonu 1 + x + x²/2! + x³/3! + x⁴/4! + ... şeklinde açılır.","Sıfır noktasındaki açılıma bakıldığında Taylor serisi, McLaurin serisi olarak adlandırılır."]},"beginTime":606,"endTime":747,"href":"/video/preview/2771589458278181586?parent-reqid=1769778737582064-10932758592689942081-balancer-l7leveler-kubr-yp-sas-212-BAL&text=0X_InRey&t=606&ask_summarization=1"},{"index":6,"title":"e üzeri ix ve Euler Formülü","list":{"type":"unordered","items":["e üzeri ix ifadesinde, x yerine ix yazarak Taylor serisi açılımı yapılır.","İmajiner birim i'nin kuvvetleri kullanılarak, e üzeri ix ifadesi reel ve imajiner kısımlara ayrılır.","e üzeri ix ifadesi, kosinüs ve sinüs fonksiyonlarıyla ilişkilendirilir: e^(ix) = cos(x) + i*sin(x)."]},"beginTime":747,"endTime":939,"href":"/video/preview/2771589458278181586?parent-reqid=1769778737582064-10932758592689942081-balancer-l7leveler-kubr-yp-sas-212-BAL&text=0X_InRey&t=747&ask_summarization=1"},{"index":7,"title":"Taylor Serisi ve Kosinüs Açılımı","list":{"type":"unordered","items":["Taylor serisi yöntemiyle cosinüs fonksiyonunun açılımı inceleniyor.","Kosinüs fonksiyonunun türevleri döngüsel olarak cosx, -sinx, -cosx, sinx şeklinde devam ediyor ve dördüncü türevi kendisine eşit.","Kosinüs x fonksiyonu Taylor serisi açılımıyla 1 - x²/2! + x⁴/4! - x⁶/6! + ... şeklinde ifade edilebilir."]},"beginTime":942,"endTime":1065,"href":"/video/preview/2771589458278181586?parent-reqid=1769778737582064-10932758592689942081-balancer-l7leveler-kubr-yp-sas-212-BAL&text=0X_InRey&t=942&ask_summarization=1"},{"index":8,"title":"Sinüs Fonksiyonunun Taylor Serisi Açılımı","list":{"type":"unordered","items":["Sinüs fonksiyonunun Taylor serisi açılımı x - x³/3! + x⁵/5! - x⁷/7! + ... şeklinde bulunuyor.","Kosinüs ve sinüs fonksiyonlarının açılımları, e^(ix) ifadesinin reel ve imajiner kısımları olarak karşılık geliyor.","e^(ix) = cosx + i sinx formülü elde ediliyor."]},"beginTime":1065,"endTime":1172,"href":"/video/preview/2771589458278181586?parent-reqid=1769778737582064-10932758592689942081-balancer-l7leveler-kubr-yp-sas-212-BAL&text=0X_InRey&t=1065&ask_summarization=1"},{"index":9,"title":"Euler Formülü ve Fiziksel Yorum","list":{"type":"unordered","items":["x yerine π koyulduğunda e^(iπ) = -1 formülü elde ediliyor.","e^(iθ) ifadesi bir karmaşık sayı olarak düşünülebilir ve bu sayı bir vektörü temsil ediyor.","θ açısı π olduğunda, boyu 1 olan vektör negatif reel ekseninde -1 noktasını gösteriyor."]},"beginTime":1172,"endTime":1349,"href":"/video/preview/2771589458278181586?parent-reqid=1769778737582064-10932758592689942081-balancer-l7leveler-kubr-yp-sas-212-BAL&text=0X_InRey&t=1172&ask_summarization=1"}],"linkTemplate":"/video/preview/2771589458278181586?parent-reqid=1769778737582064-10932758592689942081-balancer-l7leveler-kubr-yp-sas-212-BAL&text=0X_InRey&t=%%timestamp%%&ask_summarization=1"},"isAdultDoc":false,"relatedParams":{"text":"Matematiğin En Güzel Denklemi (e üzeri i.pi+1=0) 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