3个回答
求y对x的二阶导数仍然可以看作是参数方程确定的函数的求导方法，因变量由y换作dy/dx，自变量还是x，所以
y对x的二阶导数 ＝ dy/dx对t的导数 ÷ x对t的导数
dy/dt＝1/(1+t^2)
dx/dt＝1－2t/(1+t^2)＝(1+t^2-2t)/(1+t^2)
所以，dy/dx＝1/(1+t^2-2t)
d(dy/dx)/dt＝[1/(1+t^2-2t)]'＝－(2t-2)/(1+t^2-2t))^2
所以，
d2y/dx2＝d(dy/dx)/dt ÷ dx/dt
＝-(2t-2)/(1+t^2-2t))^2 ÷ (1+t^2-2t)/(1+t^2)
＝(2-2t)(1+t^2)/(1+t^2-2t)^3
y'=dy/dx=(1-2t/(1+t^2))/(1/(1+t^2))
=(t^2-2t+1)
dy'/dx=(2t-2)/(1-2t/(1+t^2))
=2(t-1)/((t-1)^2/(t^2+1))
=2(t^2+1)/(t-1)
x = x(t), y = y(t) => dy/dx = y'(t) / x'(t)
记 y'(t)/x'(t) = z(t), 考虑新的参量函数 x = x(t), z = z(t)
则 dz/dx = z'(t) / x'(t)
即 d²y/dx² = dz/dx = (dz/dt) * (dt/dx)
即证。