设f(x)=e^(x^2),则f(100阶导数）(0)等于多少.

Question

设f(x)=e^(x^2),则f(100阶导数）(0)等于多少.

Answer 1

f(x)=e^(x^2),
f'(x)=2xe^(x^2), f'(0)=0;
f''(x)=2(1+2x^2)e^(x^2), f''(0)=2*1;
f'''(x)=2^2(3x+2x^3)e^(x^2), f'''(0)=0;
f^(4)(x)=2^2(3+12x^2+4x^4)e^(2x), f^(4)(0)=2^2*3*1;
f^(5)(x)=2^3(15x+20x^3+4x^4)e^(x^2), f^(5)(0)=0;
f^(6)(x)=2^3(15+90x^2+60x^4+8x^6)e^(x^2), f^(6)(0)=2^3*5*3*1;
.
f^(n)(0)=0 n为奇数
f^(n)(0)=2^(n/2)*(n-1)!, n为偶数
故 f(100阶导数)(0) = 2^50*99!
其中 双阶乘 99!= 99*97*95*...*3*1