求解数学题：f(x)=x^5*exp(x^2)show f'(x)=2007!/1001!一道微分证明题, 急!prove f(2_其他解答

求解数学题：f(x)=x^5*exp(x^2)show f'(x)=2007!/1001!
一道微分证明题, 急!
prove f(2007)(0)=2007!/1001!not f'(x)=2007!/1001!
i.e. the 2007th derivative of f(x) equals 2007!/1001! when x=0

人气：469 ℃　时间：2020-05-11 03:10:25

解答

f(x)=x^5*exp(x^2)
用泰勒级数展开exp(x^2)然后再乘以x^5
exp(x^2)
= 1 + x^2/1! + x^4/2! + x^6/3! + ... + x^2002/1001! + .
f(x)
=x^5*exp(x^2)
= x^5 + x^7/1! + x^9/2! + x^11/3! + ... + x^2007/1001! + .
x^2007 项的系数 = f(2007)(0)/2007! = 1/1001!
故 f(2007)(0) = 2007!/1001!