参数方程如何求二阶导数？

Question

有一题目：已知参数方程：x=arctant , y= 1- ln( 1+ t ² )

我求出它的导数是 -2t , 但是不会它的求二阶导数。

求详细解答说明。高数现在我学到高阶导数。

Answer 1

dy/dx = (dy/dt)/(dx/dt) = g(t)
d²y/dx² = d/dx (dy/dx) = d/dx (g(t)) = dg(t)/dt • dt/dx = dg(t)/dt • 1/(dx/dt)

x = arctant，y = 1 - ln(1 + t²)
dx/dt = 1/(1 + t²)，dy/dx = - 2t/(1 + t²)
dy/dx = (dy/dt)/(dx/dt) = - 2t
d²y/dx² = d(dy/dx)/dx = d(- 2t)/dx = - 2 • dt/dx = - 2/(dx/dt) = - 2/[1/(1 + t²)] = - 2(1 + t²)

不妨验证下：
x = arctant ==> t = tanx
y = 1 - ln(1 + t²) = 1 - ln(1 + tan²x)
dy/dx = - 2tanxsec²x/(1 + tan²x) = - 2tanxsec²x/sec²x = - 2tanx = - 2t
d²y/dx² = d/dx (- 2tanx) = - 2 • sec²x = - 2(1 + tan²x) = - 2(1 + t²)