根据导数的定义...
lim(m->0)[f(x+m)g(x+m)-f(x)g(x)]/m
=lim[f(x+m)g(x+m)+f(x)g(x+m)-f(x)g(x+m)-f(x)g(x)]/m
=lim{g(x+m)[f(x+m)-f(x)]+f(x)[g(x+m)-g(x)]}/m
=lim g(x+m)[f(x+m)-f(x)]/m+lim f(x)[g(x+m)-g(x)]/m
=lim [f(x+m)-f(x)]/m*lim g(x+m)+lim [g(x+m)-g(x)]/m*lim f(x)
=f'(x)g(x)+f(x)g'(x)
其中
lim [f(x+m)-f(x)]/m=f'(x)
lim [g(x+m)-g(x)]/m=g'(x)
lim g(x+m)=g(x)
lim f(x)=f(x)