高数题设x=(t+1)e^t,y=t^2*e^t,求d^2y/dx^2

参数方程求导：d^2y/dx^2=d[dy/dx] / dx=d[(dy/dt) / (dx/dt)] / dx=d[y'/x']/dt * dt/dx=(y''x'-y'x'')/x'^2 * 1/x'=(y''x'-y'x'')/x'^3因此,y=(t+1)e^ty'=e^t+(t+1)e^t=(t+2)e^ty''=e^t+(t+2)e^t=(t+3)e^tx=t^2e^tx'=2te^t+t^2e^t=(t^2+2t)e^tx''=(2t+2)e^t+(t^2+2t)e^t=(t^2+4t+2)e^t代入,得：d^2y/dx^2=(y''x'-y'x'')/x'^3=[(t+3)e^t(t^2+2t)e^t-(t+2)e^t(t^2+4t+2)e^t] / [(t^2+2t)e^t]^3=[(t+3)(t^2+2t)-(t+2)(t^2+4t+2)] / e^t(t^2+2t)^3=[(t^2+3t)-(t^2+4t+2)] / e^t(t+2)^2t^3=(-t-2) / e^t(t+2)^2t^3=(-1) / e^t(t^4+2t^3)即,d^2y/dx^2=(-1) / e^t(t^4+2t^3)当然做法不止一种,介绍一种计算上或许更简单的做法（分层）：先算：dy/dx=dy/dt / dx/dt=(t+2)e^t / (t^2+2t)e^t=1/t再算：d^2y/dx^2=d(dy/dx) / dx=d(1/t)/dt / dx/dt=-1/t^2 / (t^2+2t)e^t=(-1) / e^t(t^4+2t^3)有不懂欢迎追问