f(x+y)=f(x)+f(y)+xy(x+y),①
令x=y=0得f(0)=2f(0),f(0)=0.
令y=-x,得f（0)=f(x)+f(-x),
∴f(x)是奇函数.
令y=x,得f(2x)=2f(x)+2x^3,
猜f(x)=ax^3+bx,
8ax^3+2bx=(2a+2)x^3+2bx,
比较系数得8a=2a+2,a=1/3,
f(x)=(1/3)x^3+bx,
f'(x)=x^2+b,
f'(0)=b=1.
∴f(x)=(1/3)x^3+x.
检验：f(x+y)=(1/3)(x+y)^3+x+y
=(1/3)x^3+x+(1/3)y^3+y+xy(x+y)
=f(x)+f(y)+xy(x+y).
f1(x)=(1/3)x^3+x.
下面证明本题仅此一设f(x）=f1(x)+g(x),由①,
f1(x+y)+g(x+y)=f1(x)+g(x)+f1(y)+g(y)+xy(x+y),
∴g(x+y)=g(x)+g(y),
∴g(x)=kx(k为常数）,f(x)=f1(x)+kx,
f'(x)=f1'(x)+k,
令x=0得1=1+k,k=0,
∴f(x)=f1(x).