题目

设y=y(x)由方程组x=3t^2+2t+3,e^ysint-y+1=0所确定,求当t=0时,求y对x的二阶导数

提问时间：2020-11-13

答案

x=3t^2+2t+3方程两边对t求导
dx/dt = 6t+2
e^ysint-y+1=0方程两边对t求导
e^y * (dy/dt * sint + cost) - dy/dt = 0
整理得
dy/dt=e^y * cost / (1 - e^y * sint) = e^y * cost / (2 - y)
所以根据参数方程的求导公式
dy/dx = (dy/dt) / (dx/dt) = e^y * cost / [(6t+2)(2-y)]
用对数求导法
先求对数
ln(dy/dx) = y + lncost - ln(6t+2) - ln(2-y)
对t求导
d(dy/dx)/dt / (dy/dx) = dy/dt - tant - 6/(6t+2) + （dy/dt)/(2-y)
代入数据t=0
e^ysint-y+1=0可得y=1
dx/dt = 6t+2 = 2
dy/dt=e^y * cost / (2 - y) = e
dy/dx = e^y * cost / [(6t+2)(2-y)]=e/2
d(dy/dx)/dt = (dy/dx)[dy/dt - tant - 6/(6t+2) + （dy/dt)/(2-y)] = e(2e-3)/2
所以d2y/dx2=d(dy/dx)/dt / dx/dt = e(2e-3)/4