设f(x)对任意的实数x1,x2有f(x1+x2)=f(x1)f(x2),且f'(0)=1,证明f'(x)=f(x)
问题描述:
设f(x)对任意的实数x1,x2有f(x1+x2)=f(x1)f(x2),且f'(0)=1,证明f'(x)=f(x)
答
证明:(i)设f(x)在定义域内恒不为零,由原式得:|f(x+y)|=|f(x)|*|f(y)|从而:ln|f(x+y)|=ln|f(x)|+ln|f(y)|等式两边同时对y求导得:(x+y)'f'(x+y)/f(x+y)=f'(y)/f(y)+0移项整理:f'(x+y)=f(x+y)f'(y)/f(y)=f'(y)f(...