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!

The definition will do. $$f'({\bf a},c{\bf u}) = \lim_{t \to 0} \frac{f({\bf a} + tc{\bf u}) - f({\bf a})}{t} = c \lim_{t \to 0} \frac{f({\bf a} + ct{\bf u}) - f({\bf a})}{ct} = c f'({\bf a},{\bf u})$$
Notice that $t \to 0 \iff ct \to 0$ and recognize the definition of $f'({\bf a}, {\bf u})$ in the calculation above. 
