Derivative respect to a unit vector

Question

What is derivative of $f(\hat{n},\vec{r})$ respect to the unit vector $\hat{n}=(cos(\phi),sin(\phi))$ ? $\phi$ is a unit vector in x_y plane.

Answer 1

If you are asking by the directional derivative of $f$ in the direction of $\hat{n}$ in the point $\vec{r}$, is defined as $$ \frac{\partial f}{\partial\hat{n}}(\vec{r}) = \lim_{h\to 0}\frac{f(\vec{r} + h\hat{n})}{h}. $$