My question is about the properties of directional derivatives. I came across this problem while reading this, a page on khan academy. It states that if $v^{\to}=\hat{\textbf{i}}+\hat{\textbf{j}}$, then $\nabla_{v^{\to}}f=\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}$, which I think implies that $\nabla_{v^{\to}}f=\nabla_{\hat{\textbf{i}}}f+\nabla_{\hat{\textbf{j}}}f$.
My question is, is it always true that $\nabla_{a^{\to}+b^{\to}}f=\nabla_{a^{\to}}f+\nabla_{b^{\to}}f$, and can you prove it?
Yes, linearity of the directional derivative always holds (as long as your function is differentiable - which is much stronger than having all directional derivatives) and one can prove it: see the answer here.