求n阶导数
\u6c42n\u9636\u5bfc\u6570n\u9636\u5bfc\u6570\u57fa\u672c\u90fd\u662f\u627e\u89c4\u5f8b\u3002
1\u3001y'=lnx+1\uff0cy''=x^(-1)\uff0c\u5219y\u5bf9x\u7684n\u9636\u5bfc\u6570=(-1)*(-2)*...*(-(n-2))x^(-(n-1)
2\u3001y'=e^x+x*e^x\uff0cy''=e^x+e^x+x*e^x=2e^x+x*e^x\uff0c\u5219y\u5bf9x\u7684n\u9636\u5bfc\u6570=ne^x+xe^x=(n+x)e^x\u3002
y=x^4/(x-1)
=(x^4-1+1)/(x-1)
=(x+1)(x^2+1) +1/(x-1)
=x^3 +x^2+ x+1 +1/(x-1)
\u6240\u4ee5
y'=3x^2 +2x +1 -1/(x-1)^2
y"=6x +2 +2/(x-1)^3
y"'=6 - 6/(x-1)^4
\u2026\u2026
y(n)= n! *(-1)^n / (x-1)^(n+1)\uff0cn\u5927\u4e8e\u7b49\u4e8e4
那么求导得到y'= lna *a^x
进一步求导得到n阶导数为
y(n)=(lna)^n *a^x
y=x^2 *e^2x
那么由莱布尼茨公式可以得到，
n阶导数
y(n)= (e^2x)(n) *x^2 + n *(e^2x)(n-1) *(x^2)' +n*(n-1)/2 *(e^2x)(n-2)(x^2)"+……
而x^2的2阶以上导数实际上均为常数0
e^2x的n阶导数为2^n *e^2x
所以得到y的n阶导数为
y(n)= 2^n *e^2x *x^2 + n *2^(n-1) *e^2x *(2x) + n*(n-1)/2 *2^(n-2) *e^2x *2
=2^(n-2) *e^2x *(4x^2 +4nx +n^2-n)
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