If directional derivatives in two non parallel directions are given, find the partial derivatives.

Question

If directional derivatives in two non parallel directions are given, find the partial derivatives.

First of all is it possible to find that.

All Ideas will be appreciated, thanks

Answer 1

Yes, you can always do this for a function $f: \mathbb{R}^2 \to \mathbb{R}$, provided that the function is differentiable (NB this is not the same thing as the existence of directional derivatives in nonparallel direction). The idea is that partial derivatives are just directional derivatives along the coordinate axes (in the positive directions): Thus we can write the coordinate vectors $\partial_x$ and $\partial_y$ as linear combinations of the vectors for which we know the directional derivatives. Since the derivative is linear, the partial derivatives are going to be the corresponding linear combination of the directional derivatives.

Put another way, this just amounts to computing a change of basis of the tangent space to $\mathbb{R}^2$ at a point $(x_0, y_0)$. (But, if this description doesn't feel helpful, ignore it.)