directional derivatives implications Suppose $f:\mathbb{R}^n \rightarrow \mathbb{R}^m$. Let $y\in\mathbb{R}^n$. Then we define the directional derivative in the direction $y$ by :
\begin{equation}
f'(x;y) = \lim_{t \to0}\frac{f(x+ty)-f(x)}{t} 
\end{equation}
I guess the directional derivative of $f$ in the direction of $\lambda y, \lambda\in \mathbb{R}$ should be the same with the one in the direction of $y$. But if $f'(x;y) $ exists then one can see that,
\begin{equation}
\frac{f(x+t\lambda y)-f(x)}{ t}=\lambda\frac{f(x+t\lambda y)-f(x)}{\lambda t} 
\end{equation}
and the limit of the right hand side when $t\rightarrow 0$ is $\lambda f'(x;y)$. Do I do something wrong here? Please help by giving a rigorous answer. Thank you.

You have discovered that the directional derivative, as a function of the direction, is an homogeneous function. Indeed, if $f$ is a differentiable function, the directional derivative is a linear function; in your notation: $$f'(x,\lambda v + \mu w)=\lambda f'(x,v)+\mu f'(x,w).$$
This is the chain rule applied to the function: $$g(t)=f(x+t(v+w)).$$
