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Консультация № 159611
04.02.2009, 16:36
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Помогите найти производные dy/dx данных функций: а) y=(3+6x)/корень квадратный из5x в квадрате-4х+3; б) y=arcsin корень квадратный из 1-3х; в) y=x в степени -tg x; г) y/x=arctg(x/y).
Обсуждение
Неизвестный
05.02.2009, 10:14
общий
это ответ
Здравствуйте, Попова Елена Сергеевна!
1) y=(3+6x)/(5x2-4x+3)1/2
dy/dx=[6*(5x2-4x+3)1/2 - (3+6x)*(10x-4)/(2*(5x2-4x+3)1/2)]/(5x2-4x+3)=
=[12(5x2-4x+3)-(30x+60x2-12-24x)]/(2*(5x2-4x+3)3/2)=[-30x2-34x+48]/(2*(5x2-4x+3)3/2)=
=[-15x2-17x+24]/(5x2-4x+3)3/2
2) y=arcsin((1-3x)1/2)
dy/dx=1/(1-(1-3x))1/2 * 1/(2*(1-3x)1/2) * (-3)=-3/(2sqrt(3x*(1-3x)))
3) y=x-tgx
dy/dx=x-tgx * (-tgx*lnx)'=x-tgx * (-lnx/cos2 -tgx /x)
4) y/x=arctg(x/y)
F(x,y)=y/x - arctg(x/y)=0
F'x=-y/x2 - 1/(y*((x/y)2+1)) = -y3/(x2*(x2+y2))
F'y=1/x - 1(x2/y2 +1) *(-x/y2)=(2x2+y2)/(x*(x2+y2))
dy/dx=-F'x/F'y
dy/dx=[y3/(x2*(x2+y2))]/[(2x2+y2)/(x*(x2+y2))]=y3/(x*(2x2+y2))
1) y=(3+6x)/(5x2-4x+3)1/2
dy/dx=[6*(5x2-4x+3)1/2 - (3+6x)*(10x-4)/(2*(5x2-4x+3)1/2)]/(5x2-4x+3)=
=[12(5x2-4x+3)-(30x+60x2-12-24x)]/(2*(5x2-4x+3)3/2)=[-30x2-34x+48]/(2*(5x2-4x+3)3/2)=
=[-15x2-17x+24]/(5x2-4x+3)3/2
2) y=arcsin((1-3x)1/2)
dy/dx=1/(1-(1-3x))1/2 * 1/(2*(1-3x)1/2) * (-3)=-3/(2sqrt(3x*(1-3x)))
3) y=x-tgx
dy/dx=x-tgx * (-tgx*lnx)'=x-tgx * (-lnx/cos2 -tgx /x)
4) y/x=arctg(x/y)
F(x,y)=y/x - arctg(x/y)=0
F'x=-y/x2 - 1/(y*((x/y)2+1)) = -y3/(x2*(x2+y2))
F'y=1/x - 1(x2/y2 +1) *(-x/y2)=(2x2+y2)/(x*(x2+y2))
dy/dx=-F'x/F'y
dy/dx=[y3/(x2*(x2+y2))]/[(2x2+y2)/(x*(x2+y2))]=y3/(x*(2x2+y2))
Форма ответа
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