real analysis - Counting a Directional derivative.

Let f be a function that satysfies : $$f: \mathbb{R}^2 \rightarrow \mathbb{R} , (x,y) \mapsto
\begin{cases}
0 & \text{for } (x,y)=(0,0) \\
\frac{x^3}{x^2+y^2} & \text{for } (x,y) \neq (0,0)
\end{cases} $$ I want to show that f has directional derivative at point $(0,0)$ at every direction. So, let $v=(v_1,v_2) \in R^2, (v_1,v_2) \neq (0,0)$. f has directional derivative in $(0,0)$ if and only if following limit exist :
$$\lim_{t->0}\frac{f(a+tv)-f(a)}{t}=\lim_{t->0}\frac{f(tv_1,tv_2)}{t}=\lim_{t->0}\frac{t^3v_1^3}{t(t^2v_1^2+t^2v_2^2)}=\frac{v_1^3}{v_1^2+v_2^2} $$.

So i show that above limit exist and it depends of choosing vector v. So the directional direvative exist at point $(0,0)$ in every direction. The main question is :

Am i thinking correcly ?

Answer

Yes, you are right. The directional derivative exists at $(0,0)$ in every dirction $v=(v_1,v_2)$ and is $=\frac{v_1^3}{v_1^2+v_2^2}.$