已知参数方程x=2t-t^2y=3t-t^3,求d^2y/dx^2

Question

已知参数方程x=2t-t^2 y=3t-t^3,求d^2y/dx^2

Answer 1

对于参数方程：
y=y(t)=3t-t^3
x=x(t)=2t-t^2
一阶导数：
dy/dx
=(dy/dt)/(dx/dt)
=y'/x'
=3(1-t^2) / 2(1-t)
=3(1+t)/2
那么,一阶微分：dy=3(1+t)/2 dx
二阶导数：
d^2/dx^2
=d(dy/dx)/dx
=d(y'/x')/dx
=[d(y'/x')/dt] / [dx/dt]
=[(y''x'-y'x'')/x'^2] / [x']
=(y''x'-y'x'') / x'^3
=[(-6t)(2-2t)-(3-3t^2)(-2)] / (2-2t)^3
=(-12t+12t^2+6-6t^2) / 8(1-t)^3
=(6t^2-12t+6) / 8(1-t)^3
=6(1-t)^2 / 8(1-t)^3
=3/[4(1-t)]
那么,二阶微分：d^2y=3/[4(1-t)] dx^2
我是找到复制的,不用谢我了!