This is the full question I got.
There are two unit vectors $u$ and $v$. The two vectors are not parallel. We assume that we know the directional derivatives of function $f(x, y)$ at point $(x_0, y_0)$ heading to the direction of $u$ and $v$. At this point $(x_0, y_0)$, is it possible to find the gradient $\nabla f$? If so prove it. If not, find a counterexample.
I think the answer to this question is "yes". Because the directional derivative can be written as $$ \tag{*} D_u f(x_0, y_0) = \nabla f(x_0, y_0) \cdot u $$ And I think the proof of (*) will justify my answer. Am I on the right track? Thanks in advance.