题目

求下列参数方程所确定的函数y的二阶导数d^2y/dx^2
求下列参数方程所确定的函数y的二阶导数d^2y/dx^2
1.x=2t-t^2,y=3t-t^3
2.x=f'(t),y=tf'(t)-f(t) (f''(t)≠0)

提问时间：2020-07-27

答案

1 dy/dt=3-3t^2; dx/dt=2-2t; dt/dx=1/(2-2t)
d^2y/dx^2=d(dy/dx))/dx
=[d(dy/dt * dt/dx)]/dt * dt/dx
=d[(3-3t^2)/(2-2t)]/dt * 1/(2-2t)
=3/[4(1-t)]
2 dy/dt=tf''(t);dx/dt=f''(t);dt/dx=1/f''(t)
d^2y/dx^2=d(dy/dx))/dx
=[d(dy/dt * dt/dx)]/dt * dt/dx
=d[(tf''(t))/f''(t)]/dt * 1/f''(t)
=1/f''(t)