Find the directional derivative $∂_uf(0, 0)$ for an arbitrary vector $u = (u_1, u_2) ∈\mathbb{R^2}\setminus$ $\{(0, 0)\}.$

Question

Let $f : \mathbb{R^2} \rightarrow \mathbb{R}$ be given by $$f(x, y) = \left\{ \begin{array}{c l} \frac{xy}{x+y}, & if&x+y\neq0 \\ 0, & if& x+y=0 \end{array}\right.$$

Find the directional derivative $∂_uf(0, 0)$ for an arbitrary vector $u = (u_1, u_2) ∈\mathbb{R^2}\setminus$ $\{(0, 0)\}.$

Im really not sure how to answer this question so any help will be appreciated.

Answer 1

The directional derivative at $0$ with direction $u$ is, by definition, $\lim_{h \to 0} \dfrac{f(hu) - f(0)}{h} = g'(0),$ where $g(h) = f(h u).$ So, make the substitution of $(x, y)$ by $(hu_1, hu_2)$ in the definition of $f$ and find standard derivative.