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Консультация № 161037
20.02.2009, 21:55
0.00 руб.
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3
1
Найти производную второго порядка, заданную неявно y=cos(x+y)
Обсуждение
Неизвестный
20.02.2009, 23:33
общий
Здравствуйте, Истомина Елена Андреевна!
F(x;y)=y-cos(x+y)=0 - найдём сначала частные производные 1 и 2 порядка .
dF/dx=sin(x+y) ; dF/dy=1+sin(x+y) ; ((d^2)F)/(d(x^2))=cos(x+y) ; ((d^2)F)/(d(y^2))=cos(x+y) ; ((d^2)F)/(dxdy)=cos(x+y) .
Саму формулу , извините , не помню . Довольно сложная . Есть ещё вариант выразить х через у и найти обратную производную y" .
F(x;y)=y-cos(x+y)=0 - найдём сначала частные производные 1 и 2 порядка .
dF/dx=sin(x+y) ; dF/dy=1+sin(x+y) ; ((d^2)F)/(d(x^2))=cos(x+y) ; ((d^2)F)/(d(y^2))=cos(x+y) ; ((d^2)F)/(dxdy)=cos(x+y) .
Саму формулу , извините , не помню . Довольно сложная . Есть ещё вариант выразить х через у и найти обратную производную y" .
Неизвестный
21.02.2009, 17:03
общий
это ответ
Здравствуйте, Истомина Елена Андреевна!
Берём производную по икс, считая y функцией от x.
y'=-sin(x+y)*(1+y')
Отсюда можно выразить y', он нам далее пригодится.
Теперь ещё раз берём производную:
y''=-cos(x+y)*(1+y')^2-sin(x+y)*y''
Выразите отсюда y'' и подставьте y'.
Берём производную по икс, считая y функцией от x.
y'=-sin(x+y)*(1+y')
Отсюда можно выразить y', он нам далее пригодится.
Теперь ещё раз берём производную:
y''=-cos(x+y)*(1+y')^2-sin(x+y)*y''
Выразите отсюда y'' и подставьте y'.
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