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Логаритъм: Разлика между версии

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=== Производна и антипроизводна ===
[[Файл:Logarithm derivative.svg|мини|Графика на естествен логаритъм (в зелено) и неговата тангента в {{math|''x'' {{=}} 1.5}} (в черно)|alt=Графика на логаритмична функция и нейната тангента в една точка]]
{{раздел-мъниче}}▼
 
Аналитичните свойства на функциите се предават на техните обратни функции.{{hrf|Lang|1997|}} Така, тъй като {{math|1=''f''(''x'') = {{mvar|b}}<sup>''x''</sup>}} е непрекъсната и [[диференцируема функция]], такава е и {{math|log<sub>''b''</sub>''y''}}. Грубо казано, дадена непрекъсната функция е диференцируема, ако графиката ѝ няма остри чупки. Освен това, тъй като производната на {{math|''f''(''x'')}} е равна на {{math|ln(''b'')''b''<sup>''x''</sup>}} от свойствата на експоненциалната функция, от [[верижно правило|верижното правило]] следва, че производната на {{math|log<sub>''b''</sub>''x''}} се получава като:{{hrf|Lang|1997|}}{{hrf|Wolfram Alpha|2019a}}
 
: <math>\frac{d}{dx} \log_b x = \frac{1}{x\ln b}. </math>
 
Това означава, че наклонът на [[тангента]]та към графиката на логаритъм с основа {{math|''b''}} в точката {{math|(''x'', log<sub>''b''</sub>(''x''))}} е равен на {{math|1/(''x'' ln(''b''))}}.
 
Производната на ln {{mvar|x}} е 1/''x'', от което следва, че ln {{mvar|x}} е единствената [[антипроизводна]] на {{math|1/''x''}}, която има стойност 0 за {{math|1=''x'' =1}}. Точно тази много проста формула е причина функцията да бъде наречена „естествен логаритъм“. Това е и една от основните причина за важността на константата {{mvar|e}}.
 
Производната при обобщен функционен аргумент {{math|''f''(''x'')}} е
 
:<math>\frac{d}{dx} \ln f(x) = \frac{f'(x)}{f(x)}.</math>
 
Частното вдясно се нарича [[логаритмична производна]] на ''f'', изчисляването на {{math|''f<nowiki>'</nowiki>''(''x'')}} чрез производната на {{math|ln(''f''(''x''))}} и известно като [[логаритмично диференциране]].{{hrf|Kline|1998|386}}
 
Антипроизводната на естествения логаритъм {{math|ln(''x'')}} е:{{hrf|Wolfram Alpha|2019b}}
 
: <math>\int \ln(x) \,dx = x \ln(x) - x + C.</math>
 
Подобни формули могат да се изведат от това уравнение за антипроизводните на логаритмите с друга база, като се използва правилото за промяна на основата.{{hrf|Abramowitz|1972|69}}
 
=== Интегрално представяне на естествения логаритъм ===
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=== Трансцендентност ===
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== Изчисляване ==
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== Приложения ==
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== Обобщения ==
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== Вижте също ==
Взето от „https://bg.wikipedia.org/wiki/Логаритъм“.