Question

求解数学题：f(x)=x^5*exp(x^2) show f'(x)=2007!/1001!

Answer 1

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!