Question

用微分求参数方程x=t-arctanx,y=ln(1+t²)所确定的函数y=y（x）的一阶导数

用微分求参数方程x=t-arctanx,y=ln(1+t²)所确定的函数y=y（x）的一阶导数和二阶导数

Answer 1

解：
dy/dx=(dy/dt)/(dx/dt)
=[ln(1+t²)]'/(t-arctanx)'
=[2t/(1+t²)]/[1- 1/(1+t²)]
=2t/(1+t²-1)
=2t/t²
=2/t
d²y/dx²=[d(2/t)/dt]/(dx/dt)
=(-2/t²)/[1- 1/(1+t²)]
=(-2/t²)/[(1+t²-1)/(1+t²)]
=(-2/t²)/[t²/(1+t²)]
=(-2/t²)[(1+t²)/t²]
=-2(1+t²)/t⁴