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Консультация № 202805
15.05.2022, 14:13
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Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос:
Вычислить значение производной сложной функции u=u(x,y) , где x=x(t), y=y(t) при t=t0 с точностью до двух знаков после запятой.
u=x^y, x=e^t, y=ln(t), t0=1
Вычислить значение производной сложной функции u=u(x,y) , где x=x(t), y=y(t) при t=t0 с точностью до двух знаков после запятой.
u=x^y, x=e^t, y=ln(t), t0=1
Обсуждение
16.05.2022, 11:55
общий
Адресаты:
Условие: x(t) = et , y(t) = ln(t) , u = xy .
Вычислить значение производной сложной функции u = u(x,y) при t0 = 1 .
Решение: Максимально упрощаем данную сложную функцию:
u(t) = x(t)y(t) = (et)^[ln(t)] = e^[t·ln(t)] = e^[ln(tt)] = tt , используя школьные формулы (an)k = a^(n·k) , n·ln(b) = ln(bn) , e^[ln(b)] = b
Для вычисления производной степенно-показательной функции u(t) = tt используем формулу
u'(t) = [ln(t) + 1]·tt , подробно см абзац "Производная степенно-показательной функии" учебной статьи "Сложные производные. Логарифмическая производная. Производная степенно-показательной функции" Ссылка.
При t = t0 = 1 получаем Ответ: u'(1) = [ln(1) + 1]·(11) = (0 + 1)·1 = 1
Вычислить значение производной сложной функции u = u(x,y) при t0 = 1 .
Решение: Максимально упрощаем данную сложную функцию:
u(t) = x(t)y(t) = (et)^[ln(t)] = e^[t·ln(t)] = e^[ln(tt)] = tt , используя школьные формулы (an)k = a^(n·k) , n·ln(b) = ln(bn) , e^[ln(b)] = b
Для вычисления производной степенно-показательной функции u(t) = tt используем формулу
u'(t) = [ln(t) + 1]·tt , подробно см абзац "Производная степенно-показательной функии" учебной статьи "Сложные производные. Логарифмическая производная. Производная степенно-показательной функции" Ссылка.
При t = t0 = 1 получаем Ответ: u'(1) = [ln(1) + 1]·(11) = (0 + 1)·1 = 1
Форма ответа
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