求函数的二阶导数d²y/dx². (1)x=1-t²,y=t-t³； (2)x=ln(1+t²),y=t-arctant.
(1).dy/dx=(dy/dt)/(dx/dt)=(1-3t²)/(-2t)=(3t²-1)/2t
d²y/dx²=(dy′/dt)/(dx/dt)={[(12t²-2(3t²-1)]/4t²}/(-2t)=[(6t²+2)/4t²]/(-2t)=-(3t²+1)/4t³
(2). dy/dx=(dy/dt)/(dx/dt)=[1-1/(1+t²)]/[2t/(1+t²)]=t²/2t=t/2.
d²y/dx²=(dy′/dt)/(dx/dt)=(1/2)/[2t/(1+t²)]=(1+t²)/4t.