高数 导数y=(x＾3)/(x＾2-3x+2),求y的n阶导数
y=(x＾3)/(x＾2-3x+2),求y的n阶导数
y=x^3/(x^2-3x+2)=x^3/[(x-1)(x-2)]
=(x^3-1+1)/[(x-1)(x-2)]
=[(x-1)(x^2+x+1)+1]/[(x-1)(x-2)]
=[(x^2+x+1)/(x-2)]+[1/(x-1)(x-2)]
=[(x-2)^2+5x-3]/(x-2)+[1/(x-1)(x-2)]
=(x-2)+[5(x-2)+7]/(x-2)+[1/(x-1)(x-2)]
=(x-2)+5+[7/(x-2)]+[1/(x-2)]-[1/(x-1)]
=x+3+8*[1/(x-2)]-[1/(x-1)]
=x+3...全部
y=(x＾3)/(x＾2-3x+2),求y的n阶导数
y=x^3/(x^2-3x+2)=x^3/[(x-1)(x-2)]
=(x^3-1+1)/[(x-1)(x-2)]
=[(x-1)(x^2+x+1)+1]/[(x-1)(x-2)]
=[(x^2+x+1)/(x-2)]+[1/(x-1)(x-2)]
=[(x-2)^2+5x-3]/(x-2)+[1/(x-1)(x-2)]
=(x-2)+[5(x-2)+7]/(x-2)+[1/(x-1)(x-2)]
=(x-2)+5+[7/(x-2)]+[1/(x-2)]-[1/(x-1)]
=x+3+8*[1/(x-2)]-[1/(x-1)]
=x+3+8*(x-2)^(-1)-(x-1)^(-1)
所以：
y'=1-8(x-2)^(-2)+(x-1)^(-2)
y''=(-8)*(-2)*(x-2)^(-3)+(-2)*(x-1)^(-3)
=16*(x-2)^(-3)-2*(x-1)^(-3)
=8*2!*(x-2)^(-3)-2!*(x-1)^(-3)
y'''=8*2!*(-3)*(x-2)^(-4)-2!*(-3)*(x-1)^(-4)
=-8*3!*(x-2)^(-4)+3!*(x-1)^(-4)
……
所以：
y(n)'=(-1)^n*n!*[8*(x-2)^(-n-1)-(x-1)^(-n-1)]（n≥2）。
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