已知y=x^2*sin2x.求y的50阶导数?

Question

已知y=x^2*sin2x.求y的50阶导数?

Answer 1

(uv)'=u'v+uv'
(uv)''=((uv)')'=(u'v+uv')'= u''v+u'v'+u'v'+uv''=u''v+2u'v'+uv''
(uv)'''=(u''v+2u'v'+uv'')'=(u'''v+u''v'')+(2u''v'+2u'v'')+(uv'''+u'v'')=u'''v+3u''v'+3u'v''+uv'''
(uv)(n) = C(0,n)u(0)v(n)+C(1,n)u(1)v(n-1)+C(2,n)u(2)v(n-2)+.+C(n,n)u(n)v(0)
C(0,n),C(1,n),C(2,n)这些是排列组合,u(n),v(n)表示n阶导数
因此你的题目中
u=x^2 ,v=sin2x
u'=2x,u''=2,u'''=0,因此,u的三阶导数以上都是零了,上面的展开式只需要求前面含有的u的零阶、一阶和二阶导数的项C(0,50)u(0)v(50)、C(1,50)u(1)v(49)、C(2,50)u(2)v(48)就可以了
u(0)=x^2 u(1)=2x u(2)=2
sin(kx)(n)=k^nsin(kx+0.5nπ)
v(50)=2^50sin(2x+25π)= -2^50sin2x
v(49)=2^49sin(2x+24.5π)=2^49sin(2x+0.5π)= -2^49cos2x
v(48)=2^48sin(2x+24π)=2^48sin2x
C(0,50)=1,C(1,50)=50,C(2,50)=49*25=1225
y(50)=C(0,50)u(0)v(50) + C(1,50)u(1)v(49) + C(2,50)u(2)v(48)
=x^2·(-2^50sin2x) + 100x·(-2^49cos2x) + 1225×2^48sin2x