Prove partial derivatives exist, but not all directional derivatives exists. During my analysis course my teacher explained the difference between partial derivatives and directional derivatives using the notion that a partial derivatives looks at the function as approaching a point along the axes (in case of of the plane), and a directional derivative as approaching a point from any direction in the plane. He also explained that the existence of directional derivatives is a stronger notion than the existence of partial derivatives exists: if all directional derivatives exist, then the partial derivatives exist too.
I am to show (not necessarily prove) a case where the partial derivatives exist, but not all directional derivatives exist (hence, f is not differentiable).

Consider for instance
$$f(x,y)=\begin{cases}
\frac{xy}{x^2+y^2}&\text{if }(x,y)\neq(0,0),\\
0&\text{otherwise.}\end{cases}$$
The partial derivatives exist everywhere. Away from the origin this is clear. At the origin we need to calculate the partials from first principles:
$$\frac{f(0+h,0)-f(0,0)}h=\frac{0-0}h=0$$
so letting $h\to0$ we find $f_x(0,0)=0$. By symmetry we also have $f_y(0,0)=0$. However, not all directional derivatives exist at the origin. For example, let $\mathbf v=(\frac1{\sqrt{2}},\frac1{\sqrt{2}})$. Then
$$\frac{f(\mathbf0+t\mathbf v)-f(\mathbf0)}{t}=\frac{\frac{t^2}{t^2+t^2}-0}t=\frac1{2t}$$
which does not have a limit as $t\to0$.
