Let $\displaystyle A\subset\Re^m$ ; let $\displaystyle f:A\rightarrow\Re^n$ Show that if $\displaystyle f'(a;u)$ exists, then $\displaystyle f'(a:cu)$ exist and equals $\displaystyle cf'(a;u).$

Some people in my class were trying to say that we know that by theorem

$\displaystyle f'(a:u)=Df(a)*u$

But I disagree since we don't know that Df(a) exist but I don't know how to prove this result. Any ideas would be helpful.