f(X+Y)=f(X)f(Y),且f(0)的導數存在,求證f `(x)=f(x)f `(0)

f(X+Y)=f(X)f(Y),且f(0)的導數存在,求證f `(x)=f(x)f `(0)

根據導數定義,f'(x)=[f(x+t)-f(x)]/t(其中t是趨向於0的一個x的小分割),則f'(x)=[f(x)*f(t)-f(x)]/t=f(x)*[f(t)-1]/t由題意可知:f(x)=f(x+0)=f(x)*f(0),則有f(0)=1那麼f'(x)=f(x)*[f(0+t)-f(0)]/t=f(x)*f'(0)…