西安毛子律枝大学-例5 对任意正数x,y有 f(xy)= f(x)+ f(y)且 f'(l)=1.证明f(x)在(O,+o)内可导.证 f(11)= f(1)+ f(1), :: f(1)=0, Vxe(0,+),有f[x·(1+^)]- f(x)f(x+h)-f(x)f'(x)= lim ==lim4hhh->0h-→0hf(1+f(x)+ f(1+f(x)f(1xxx= limlimlim-hhhh-0h->0h->0xx1'(1)存在xx例5 对任意正数 x y, 有 f x y f x f y ( ) ( ) ( )  = + 且 f (1) 1, = 证明 f x( ) 在 (0, ) + 内可导. 证 f f f (1 1) (1) (1),  = +  = f (1) 0,   + x (0, ),有 0 ( ) ( ) ( ) limh f x h f x f x → h + −  = 0 [ (1 )] ( ) lim h h f x f x x → h  + − = 0 ( ) (1 ) ( ) lim h h f x f f x x → h + + − = 0 (1 ) lim h h f x → h + = 0 (1 ) (1) 1 lim h h f f x h x x → + − =  1 f (1) x =   1 . x = 存在