Question

用积分求参数方程x=t-arctant,y=ln(1+t^2)确定的函数y=y(x)的一阶导数和二阶导数.

Answer 1

dx/dt=1-1/(1+t^2)=t^2/(1+t^2)

dy/dt=2t/(1+t^2)

dy/dx=(dy/dt)/(dx/dt)=2t/t^2=2/t

同理求d^2x/dt^2=2t/(1+t^2)^2

d^2y/dt^2=(2-2t^2)/(1+t^2)^2

d^y/dx^2=(1-t^2)/t