Proof of a property of directional derivative I am stuck with the proof of the following proposition. I am given that the directional derivative of f exists at a with respect to the vector u, and I should prove that
f'(a,cu)=cf'(a,u)
I tried to use the theorem  that if f is differentiable at a, then f'(a,u) = f'(a)*u. Using this theorem, I would prove the proposition. But the condition that f is differentiable at a is not given. So I can't proceed
Any hint would be appreciated.
Thanks in advance!

$\begin{align*}
f'(a,cu) &= \lim_{h \to 0} \frac{f(a+hcu)-f(a)}{h}\\
&= \lim_{h \to 0} c\frac{f(a+hcu)-f(a)}{ch}\\
&= c\lim_{ch \to 0} \frac{f(a+hcu)-f(a)}{ch}\\
&=cf'(a,u)
\end{align*}$
