求 x=1 的三阶导数,x^2 + xy + y^3 = 1如题,f'(x) = -2,f''(x) = 2,f'''(x) = 42,我算三阶导数只得 -30,用隐函数算
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![求 x=1 的三阶导数,x^2 + xy + y^3 = 1如题,f'(x) = -2,f''(x) = 2,f'''(x) = 42,我算三阶导数只得 -30,用隐函数算](/uploads/image/z/11604445-61-5.jpg?t=%E6%B1%82+x%3D1+%E7%9A%84%E4%B8%89%E9%98%B6%E5%AF%BC%E6%95%B0%2Cx%5E2+%2B+xy+%2B+y%5E3+%3D+1%E5%A6%82%E9%A2%98%2Cf%27%28x%29+%3D+-2%2Cf%27%27%28x%29+%3D+2%2Cf%27%27%27%28x%29+%3D+42%2C%E6%88%91%E7%AE%97%E4%B8%89%E9%98%B6%E5%AF%BC%E6%95%B0%E5%8F%AA%E5%BE%97+-30%2C%E7%94%A8%E9%9A%90%E5%87%BD%E6%95%B0%E7%AE%97)
求 x=1 的三阶导数,x^2 + xy + y^3 = 1如题,f'(x) = -2,f''(x) = 2,f'''(x) = 42,我算三阶导数只得 -30,用隐函数算
求 x=1 的三阶导数,x^2 + xy + y^3 = 1
如题,f'(x) = -2,f''(x) = 2,f'''(x) = 42,我算三阶导数只得 -30,用隐函数算
求 x=1 的三阶导数,x^2 + xy + y^3 = 1如题,f'(x) = -2,f''(x) = 2,f'''(x) = 42,我算三阶导数只得 -30,用隐函数算
由x^2+xy+y^3=1 (@)
x=1时y=0
两边求导
2x+xy'+y+3y^2*y'=0(@@)
代入数据得y'=-2
对@@两边求导
2+xy''+y'+y'+3y^2*y''+6yy'^2=0
整理得
2+(x+3y^2)y''+2y'+6yy'^2=0(@@@)
代入数据得y''=2
对@@@两边求导
(x+3y^2)y'''+(1+6yy')y''+2y''+6y*2y'y''+6y'^3=0
整理得
(x+3y^2)y'''+(3+18yy')y''+6y'^3=0
代入数据得y'''=-3*(-2)-6*(-2)^3=42
当x=1时,y=0.
求导得: 2x y xy' 3y^2y'=0,代入求得y'(1)=-2.
再求导得:2 y' y' xy'' 6y(y')^2 3y^2y''=0,代入x,y,y'求得:y''(1)=2.
再求导得:2y'' y'' xy"' 6y'(y')^2 12yy'y" 6yy'y" 3y^2y"'=0,代入x,y,y',y"得:
4 2 y"'(1)-48=0,y"'(1)=42