Question

微积分高阶导数问题，在线等

求参数方程所确定的函数的二阶导数，x=f '(t)
y=t f '(t)+f(t)
其中f''(t)存在非0
注意是二阶导数，答案是1/f''(t)
求过程？咋个来的？

Answer 1

dx/dt = f''(t)
dy/dt = f'(t) + tf''(t) + f'(t) = 2f'(t) + tf''(t)
dy/dx = [2f'(t) + tf''(t)]/f''(t) = 2f'(t)/f''(t) + t
d²y/dx² = [d(dy/dx)/dt][dt/dx]
= {2f''(t)/f''(t) - 2f'(t)f'''(t)/[f''(t)]² + 1}[1/f''(t)]
= [2 - 2f'(t)f'''(t)/[f''(t)]² + 1][1/f''(t)]
= 3/f''(t) - 2f'(t)f'''(t)/[f''(t)]³

题目应该有误.

Answer 2

dy/dx=(2f'(t)+t*f''(t))/(f''(t))