高数如图图

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(5)

y=1+xe^y
y'= (1+xy').e^y
(1-xe^y)y' = e^y
y' = e^y/(1-xe^y)
y''
= e^y.[ (1-xe^y) +(1+xy') ] /(1-xe^y)^2
= e^y.( 2-xe^y+xy' ) /(1-xe^y)^2
= e^y.[ 2-xe^y+ x.e^y/(1-xe^y) ] /(1-xe^y)^2
= e^y.[ (1-xe^y)(2-xe^y)+x.e^y ] /(1-xe^y)^3
= e^y.[ 2- 2x.e^y +x^2.e^(2y) ] /(1-xe^y)^3
(7)
x=f(t)
dx/dt = f'(t)
y=t+f(t)
dy/dt = 1 +f'(t)
dy/dx = (dy/dt)/(dx/dt) = [ 1+f'(t) ]/f'(t) = 1 + 1/f'(t)
d/dt (dy/dx ) = - f''(t)/[f'(t)]^2
d^2y/dx^2
=d/dt (dy/dx ) / (dx/dt)
= {- f''(t)/[f'(t)]^2 }/ f'(t)
=- f''(t)/ [f'(t)]^3