函数y=y(x)由方程x=1+2t^2,y=∫(e^u)/u du (t>1)确定,求x=9时,d^2y/dx^2
问题描述：
函数y=y(x)由方程x=1+2t^2,y=∫(e^u)/u du (t>1)确定,求x=9时,d^2y/dx^2
其中y=∫(e^u)/u du (t>1)是从1积到1+2lnt的定积分
答
dy/dt = {e^(1+2lnt) ) / (1+2lnt) } 2/t = 2et / (1+2lnt)
dx/dt = 4t
dy/dx=(dy/dt ) / (dx/dt )= {2et / (1+2lnt) } / 4t =et/ (2+4lnt)
d(dy/dx) /dt ={e(2+4lnt) - 4e } / (2+4lnt)^2= e(4lnt-2) / (2+4lnt)^2
d^2y/dx^2={d(dy/dx) /dt }/ (dx/dt )
={e(4lnt-2) / (2+4lnt)^2}/4t
=e(2lnt-1) / { 8t(1+2lnt)^2}
d^2y/dx^2 ( 9 ) = e(2ln9-1) / { 72 (1+2ln9)^2}