Whenever we deal with functions such as $f:\mathbb{R}^n \to \mathbb{R}$, when we defined the notion of differentiability or the directional derivative, we use the coordinates $(1,0,...,0), (0,...,1,...,0)$ etc.. However, is it possible to use the same rules and definitions to show the differentiability of a function or find its directional derivate using any basis of $\mathbb{R}^n$ (maybe particularly only using any orthonormal basis of $\mathbb{R}^n$).
So for example, if $u, v$ is any 2 linearly independent vectors in $\mathbb{R}^2$, and $x_1, x_2$ is the standart basis of $\mathbb{R}$, does the following equality $$D_{x_1} f * dx_1 + D_{x_2} f * dx_2 = D_u f * du + D_v f * d_v$$ hold ? If so, how can we prove it ?