设函数y=y(x)由方程x^2y^2+y=1(y>0)所确定 求dy/dx|x=0,d^2y/dx^2|x=0.
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设函数y=y(x)由方程x^2y^2+y=1(y>0)所确定 求dy/dx|x=0,d^2y/dx^2|x=0.
![设函数y=y(x)由方程x^2y^2+y=1(y>0)所确定 求dy/dx|x=0,d^2y/dx^2|x=0.](/uploads/image/z/2860999-7-9.jpg?t=%E8%AE%BE%E5%87%BD%E6%95%B0y%3Dy%28x%29%E7%94%B1%E6%96%B9%E7%A8%8Bx%5E2y%5E2%2By%3D1%28y%3E0%29%E6%89%80%E7%A1%AE%E5%AE%9A+%E6%B1%82dy%2Fdx%7Cx%3D0%2Cd%5E2y%2Fdx%5E2%7Cx%3D0.)
∵x²y²+y=1(y>0)
==>2xy²+2x²yy'+y'=0 (等式两端对x求导数).(1)
==>2y²+4xyy'+4xyy'+2x²(yy''+y'²)+y''=0 (等式两端对x求导数).(2)
∴由(1)得y'=-2xy²/(2x²y+1)
由(2)得y''=-(2y²+8xyy'+2x²y'²)/(2x²+1)
∵当x=0时,y=1
∴dy/dx|(x=0)=y'|(x=0)=-2*0*1²/(2*0²*1+1)=0
d²y/dx²|(x=0)=y''|(x=0)=-(2*1²+8*0*1*0+2*0²*0²)/(2*1²+1)=-2/3.
==>2xy²+2x²yy'+y'=0 (等式两端对x求导数).(1)
==>2y²+4xyy'+4xyy'+2x²(yy''+y'²)+y''=0 (等式两端对x求导数).(2)
∴由(1)得y'=-2xy²/(2x²y+1)
由(2)得y''=-(2y²+8xyy'+2x²y'²)/(2x²+1)
∵当x=0时,y=1
∴dy/dx|(x=0)=y'|(x=0)=-2*0*1²/(2*0²*1+1)=0
d²y/dx²|(x=0)=y''|(x=0)=-(2*1²+8*0*1*0+2*0²*0²)/(2*1²+1)=-2/3.
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