求下列参数方程所确定的函数的导数1.x=e∧tsinty=e∧tcost求dy／dx2.x=根号(1+t²)y=arctant求dy／dx3.x=sinty=cos2t求d²y／dx²4.x=2e∧ty=3e∧(-t)求d²y／dx²

Question

求下列参数方程所确定的函数的导数
1.x=e∧t sint
y=e∧t cost 求dy／dx
2.x=根号(1+t²)
y=arctant 求dy／dx
3.x=sint
y=cos2t 求d²y／dx²
4.x=2 e∧t
y=3 e∧(-t) 求d²y／dx²

Answer 1

①根据原参数式可以得到
dx = coste^tdt + sinte^tdt = e^t(sint + cost)dt
dy = coste^tdt - sinte^tdt = e^t(cost - sint)dt
dy/dx = (sint + cost) / (cost - sint)
sint = xe^(-t) cost = ye^(-t)
dy/dx = (x + y)/(x - y)
②根据原参数式可以得到
x^2 = 1 + t^2
tany = t
x^2 = 1 + (tany)^2 = 1/(cosy)^2
(xcosy)^2=1
d(xcosy)^2=0
2(xcosy)(cosydx - xsinydy)=0
cosy dx = xsiny dy
dy/dx = cosy/(xsiny) = coty/x　
③根据原参数式可以得到
dx = costdt dy = -2sin2t = -4sintcostdt
dy/dx = -4sint
d^2y/dx^2 = d(dy/dx)/dx = (-4sint)'/cost = -4cost/cost = -4
④根据原参数式可以得到
dx = 2e^tdt
dy = -3e^(-t)dt
dy/dx = -1.5e^(-2t)
d(dy/dx) = 3e^(-2t)dt
d^2ydx^2 = 3e^(-2t)/2e^t = 1.5e^(-3t)