高数导数问题参数方程求二次导数时,对dy/dx再求导变成(d^2y)/dx^2=d*dy/dx^2这里d与y拆开后d代表什么?配合例题讲解下:求摆线参数方程:x=a(t-sint)y=a(1-cost)所表示的函数y=y(x)的二阶导数1-costnott-cost

Question

高数导数问题
参数方程求二次导数时,对dy/dx再求导
变成(d^2y)/dx^2=d*dy/dx^2
这里d与y拆开后d代表什么?
配合例题讲解下:
求摆线参数方程:
x=a(t-sint)
y=a(1-cost)所表示的函数y=y(x)的二阶导数
1-cost
not t-cost

Answer 1

dy/dx是一阶导数
d^2y/dx^2是二阶导数
d^2y/dx^2=dy'/dx
y'=dy/dx
x=a(t-sint)
y=a(1-cost)
一阶导数
y'=dy/dx
=da(1-cost)/da(t-sint)
=[a(1-cost)]'/[a(t-sint)]'
=asint/a(1-cost)
=sint/(1-cost)
二阶导数
y''=dy'/dx
=d(sint/(1-cost))/da(t-sint)
=[(sint/(1-cost)]'/[a(t-sint)]'
=[(cost(1-cost)-sint(sint))/(1-cost)^2]/a(1-cost)
=[(cost-(cost)^2-(sint)^2)/(1-cost)^2]/a(1-cost)
=(cost-1)/a(1-cost)^3
= -1/a(1-cost)^2
注意：楼上的dy/dt=a(1+sint) 出问题了,应该是dy/dt=asint